Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 57, pp. 1–32.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu

DOI: 10.58997/ejde.2025.57

MODELING OF GROUNDWATER FLOW IN POROUS MEDIUM LAYERED

OVER INCLINED IMPERMEABLE BEDS

PETR GIRG, LUKÁŠ KOTRLA

In memory of Professor John W. Neuberger with admiration

Abstract. We propose a new mathematical model of groundwater flow in porous medium

layered over inclined impermeable beds. In its full generality, this is a free-surface problem.

To obtain analytically tractable model, we use generalized Dupuit-Forchheimer assumption for
inclined impermeable bed. In this way, we arrive at parabolic partial differential equation which

is a generalization of the classical Boussinesq equation. The novelty of our approach consists
in considering nonlinear constitutive law of the power type. Thus introducing p-Laplacian-like

differential operator into the Boussinesq equation. Unlike in the classical case of the Boussinesq

equation, the convective term cannot be set aside from the main part of the diffusive term and
remains incorporated within it.

In this article, we analyze qualitative properties of the stationary solutions of our model.

In particular, we study the existence and regularity of weak solutions for the boundary value
problem

−
d

dx

[
(u(x) +H)|

du

dx
(x) cos(φ) + sin(φ)|p−2

(du

dx
(x) cos(φ) + sin(φ)

)]
= f(x), x ∈ (−1, 1) ,

u(−1) = u(1) = 0 ,

where p > 1, H > 0, φ ∈ (0, π/2), f ≥ 0, f ∈ L1(−1, 1). In the case of p > 2, we study validity

of Weak and Strong Maximum Principles as well. We use methods based on the linearization of
the p-Laplacian-type problems in the vicinity of known solution, error estimates, and analysis

of Green’s function of the linearized problem.

1. Introduction

Hand in hand with global warming (whether man made or not), global water cycle intensifies
and hydrological extremes (such as heavy precipitation events and local floods) may occur more
frequently, see, e.g., [33, 36, 62, 65]. Thus, further research and development of more effective
drainage systems is needed. A typical situation which frequently appears in this context is water
flow in a porous medium layered over a sloping impermeable bed. In practice, this situation can
be encountered, e.g., in highway and railway drainage [31, 68], buried streams through coarse
porous media and valley fills [35, 51, 52], water seepage through soil between parallel ditches in
irrigated/drained sloping lands [15, 16, 49, 56, 66] and/or groundwater flow in inclined phreatic
aquifers [5, 6, 7, 67]. In its full generality, this flow configuration leads to a free boundary value
problem, which is often very difficult to analyze and/or use in simulations, see, e.g., [2]. For
easier application in practice, several simplified mathematical models were proposed and studied
in a number of papers, see, e.g., [7, 15, 16, 19, 42, 66] and references therein. Most of these
models study groundwater flow in a soil layered over impermeable bed and use linear constitutive
law, known as Darcy’s law (see [20], and also, e.g., [1, 9, 10, 34, 45, 48] for further discussions).

2020 Mathematics Subject Classification. 76S05, 35Q35,34B15, 34B27.
Key words and phrases. Porous medium; filtration; nonlinear Darcy’s law; p-Laplacian;

pressure-to-velocity power law.
©2025. This work is licensed under a CC BY 4.0 license.
Submitted January 3, 2025. Published May 30, 2025.

1



2 P. GIRG, L. KOTRLA EJDE-2025/57

However, it has been established by numerous complex laboratory experiments and by many in-
field observations that the Darcy’s law is not satisfactory for certain materials and flow regimes,
see, e.g., [30, 37, 41, 43, 57, 58, 59]. However, typical materials used in modern drainage systems
comprise of coarse porous media such as gravel or geosynthetic materials and it turns out that
movement of water in such materials is more accurately described by nonlinear constitutive laws
such as power type law, sometimes called Smreker-Izbash-Missbach law, or polynomial type law,
known as Forchheimer’s law. Empirical studies on these materials can be found in [53] (gravels),
[12] (geosynthetic materials), [28] (porous asphalt), [40, 50] (railway ballast material). Moreover,
it has been experimentally established that the groundwater obeys nonlinear constitutive law of
power type in low permeability materials such as certain sandstones, fine sands, clays or certain
types of soils, see, e.g., [39, 59, 71].

The purpose of this article is twofold. At first we propose an improved mathematical model
of water flow in a porous medium obeying power law, layered over a sloping impermeable bed in
Section 2. Then we present some mathematical tools suitable for its study in Section 3. These tools
are mainly based on linearization of quasilinear operators of the p-Laplacian type. In particular,
we prove validity of Strong Maximum Principle in certain situations that can be met in analyzing
real world situations. Concluding remarks with a table summarizing results of the papers are in
Section 4. Finally, technical parts of some proof from Section 3 can be found in the Appendix 5.

We use usual notation throughout this article such as, Lp(−1, 1), p ≥ 1, stands for spaces of
Lebesgue integrable functions, C[a, b], C(a, b) with a < b stand for spaces of continuous functions
on respective intervals [a, b] and (a, b), AC[−1, 1] stands for the space of absolutely continuous
functions on [−1, 1]. Analogously, Ck[a, b] and Ck(a, b), k ∈ N, stands for spaces of k-times
continuously differentiable functions, C∞(a, b) = ∩k∈NC

k(a, b) and C∞
c (−1, 1) stands for the space

of smooth functions u with compact support suppu ⊂ (−1, 1). By W 1,p
0 (−1, 1), p > 1, we denote

the space of functions u from Lp(−1, 1) having its distributional derivatives in Lp(−1, 1) and
satisfying u(−1) = u(1) = 0 in the sense of traces. Let us note that the boundary conditions are

satisfied in the classical sense, since any u ∈ W 1,p
0 (−1, 1) has a representative in AC[−1, 1] (see,

e.g., [70, Thm. 2.1.4]). We denote the norm of a Banach space X by ∥·∥X , with the only exception
L∞(−1, 1) whose corresponding norm is denoted by ∥ · ∥∞ for brevity. Throughout the paper, we
will also use the following standard notation

u+ def
= max{u, 0} , u− def

= max{−u, 0} ,
for any real function u (recalling that u = u+ − u−).

2. Mathematical model of water flow in porous medium layered over an inclined
impermeable bed

2.1. Initial physical considerations. Generally, water tends to flow from places with higher
potential energy to places with lower potential energy. While the water is flowing, its potential
energy is transformed into kinetic energy, which is further dissipated by viscosity forces and friction
with the porous medium. Since the actual velocity of the groundwater highly oscillates in the
channels in the porous medium, one must consider bulk motion of the water within a sufficiently
large control volume in the porous medium. This averaged velocity v⃗ is defined by means of specific
discharge q⃗ by the formula v⃗ = q⃗/n, where the specific discharge (vector) q⃗ takes the direction of
the flow and its magnitude is defined as the volume of water flowing per unit time through a unit
cross-sectional area normal to the direction of flow, and n is effective porosity of the medium see,
e.g., [9, p. 121] for detailed explanation. On this macroscopic level, the process of transformation
and dissipation of energy can be described in the following way. The total mechanical energy per
volume in a control volume of water is the sum of gravitational potential energy, pressure energy,
and kinetic energy

ET = zϱg + P +
1

2
ϱv2 ,

where v stands for the magnitude of averaged velocity of the flow in the control volume, ϱ is
the water density, P pressure, z elevation of the control volume from the datum, g gravitational



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 3

acceleration, see, e.g., [54]. For incompressible liquid such as water, one can equivalently consider
another quantity called total head

hT
def
=

ET

ϱg
= z +

P

ϱg
+

1

2g
v2

which can be directly measured in practice, e.g., by using observation wells or by so called piezome-
ters, see, e.g., [9, p. 63] or [54, pp. 129–132]. As it was already mentioned, groundwater is loosing
its total energy (or equivalently total head) while flowing due to viscous forces and friction with
porous medium. Thus, its total energy decreases in the direction of the flow. The term correspond-
ing to kinetic energy is negligible and can be dropped for the average velocities of the groundwater
flow encountered in real situations, see, e.g., [34, p. 5] or [54, pp. 40–43]. In this way, we obtain
piezometric head

h
def
= z +

P

ϱg

which is the state variable in the mathematical models of the water flow in the underground. On
the other hand, the specific discharge is the flux quantity. The constitutive law relating this two
quantities, quantitatively describes the rate of dissipation of the energy along the flow path.

2.2. Constitutive law. In practice, the constitutive law is obtained empirically from experimen-
tal data for given porous medium and fluid. Linear Darcy’s law relates groundwater flux to the
piezometric head loss per length according to the following formula

v = c
△h

△L
,

where c > 0 is a constant to be determined from measured data, △h is the difference of the piezo-
metric head measured at two distinct locations distance △L apart. This formula was established
experimentally for filtration of water through sand by Henry Darcy [20] in 1856 and soon it be-
came widely popular in mathematical models of groundwater flow due to its simplicity (and still
reasonable accuracy). Unfortunately, it was later found that it has limited range of its validity in
coarse grained media (such as gravels), see, e.g., [30, 37, 41, 43, 57, 58] as well as in media with
very low permeability (such as clays, certain soils, sandstones), see, e.g., [39, 71]. For thorough
surveys and discussions of this and other constitutive laws and various criteria of their validity,
see, e.g., [1, 9, 48, 54, 59]. It follows from these discussions and the above mentioned papers that
the power type law

v = c
(△h

△L

)m

(2.1)

with constants c,m > 0 to be empirically determined, turned out to be simple but flexible enough
to fit with most experimental data obtained for various porous media. For m = 1, the power law
coincides with the Darcy’s law.

Note that the constitutive laws are inferred from experiments made on one dimensional flow
and that the averaged velocity is (practically) constant within the sample of material and during
the time of the measurement. However, groundwater flow in the real world is three dimensional,
in general, and the physical quantities v⃗ and h are usually functions of spatial variables and time.
In the case of the homogeneous and isotropic porous medium, the three-dimensional constitutive
law can be inferred from the one-dimensional one in a straightforward manner, taking into account
that the averaged velocity takes the opposite direction of the gradient of the piezometric head and
no flow occurs if the gradient of the piezometric head is zero. In this way, we obtain

v⃗ =

{
0⃗ for ∇h = 0⃗ ,

−c|∇h|m−1∇h for ∇h ̸= 0⃗ ,
(2.2)

where ∇h stands for the spatial gradient of the piezometric head, c,m > 0 are constants as in
(2.1).

Note that the power law (2.1) is not the only type of nonlinear laws used in practice. For
thorough surveys of other important types of nonlinear constitutive laws, see, e.g., [1, 9, 48, 54].



4 P. GIRG, L. KOTRLA EJDE-2025/57

Furthermore, a discussion of development of constitutive laws for fluid flows in porous media from
the perspective of history of science can be found, e.g., in [1, 10].

2.3. Free surface problem and Dupuit-Forchheimer assumption. The groundwater flow
with a free-surface upper boundary in a porous medium layered over impermeable bed is very
challenging problem, since a part of the boundary of the domain is not known a priori and thus
it is one of the unknowns. Rigorous formulation and mathematical treatment of some important
cases of problems with free-surface upper boundary can be found, e.g., in [2]. Since these types
of problems are frequently encountered in engineering, it was as early as in the middle of the
nineteenth century when French engineer J. Dupuit [27] (cf. also [26] for similar approach to flow
in open channels) published a relatively simple method how to find approximate solutions of these
type of problems and used this method to study groundwater flow with free surface towards a fully
penetrating well. His method was based on observation that the maximum slope of the upper free
surface is very small, (typically △h/△L is of order 0.001). This lead him to the following two
simplifying assumptions:

(A1) the groundwater flow is horizontal (piezometric head is constant in vertical direction),
(A2) the groundwater flow is proportional to the gradient of the piezometric head.

A

B

v⃗h

????

rain

Figure 1. Classical Dupuit-Forchheimer assumption (A1). The flow is horizontal
with the velocity distribution profile and the piezometric head being constant
along vertical line segment AB.

Since the capillarity effects are also neglected in his approach, the hydrostatic pressure must
be equal to atmospheric pressure at the free-surface. Therefore, he obtained the relation

h(x, y, z, t) = z ,

for any point (x, y, z) at the free surface at any time t, see Figure 1. It then follows from the
assumption (A1) that h(x, y, z, t) = h(x, y, t) (with a little abuse of notation) and the height z
of the free-surface boundary above the impermeable layer is then z = h(x, y, t). Thus it reduces
the original 3D problem with unknown free-surface upper boundary to a problem in the xy-plane,
where the surveyed area is known in advance, see, e.g., [1, 9, 34] for derivation of the corresponding
equations in modern notation.



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 5

Independently, similar research was performed by Austrian engineer P. Forchheimer [29], where
groundwater flows towards a system of fully penetrating wells, or towards a fully penetrating slot
of finite length were studied. The approach by Dupuit [27] and Forchheimer [30] turned out to
be very useful and enabled to solve many practical problems arising from engineering. However,
one has to bear in mind that it is only an approximation of the original problems. There is a
thorough discussion in [9, 34] of various cases, for which Dupuit-Forchheimer assumption leads to
a reasonable approximation of the free surface problem as well as the cases for which it yields a
poor approximation. Results obtained using the Dupuit-Forchheimer assumption were compared
to experimentally measured data, e.g., in [67].

In [30, 69], these ideas were extended to groundwater flow in natural coarse grained media such
as gravel, where nonlinear constitutive laws better fits experimental data. In both papers [30, 69],
authors used assumption (A1) with nonlinear power law (2.1) (there, written in another equivalent
forms) to solve a problem of groundwater flow towards a fully penetrating well, and also towards
an infinite fully penetrating ditch in case of [30].

2.4. Dupuit-Forchheimer assumption on the sloping porous medium. The situation be-
comes even more complicated when the impermeable bed (upon which the porous medium is
resting) is inclined. There are two main approaches how to adopt Dupuit-Forchheimer assump-
tion to this situation. The simpler approach is based on (A1), that is, the flow is horizontal despite
the impermeable bed is inclined. Thus it turns out that this approach is reasonable only for very
small inclinations. It was already in 1848, when Dupuit [26] introduced this approach to study
water flow in an inclined open channel and later adopted in Boussinesq [13] for the groundwater
flow in a porous medium over an inclined impermeable bed. For its simplicity, it is widely used in
solving engineering problems see, e.g., monographs [9, 34] and [5, 3, 4, 15, 49].

The second approach suggested already by Boussinesq [14, pp. 252–260] in 1877, and later
promoted by Childs [16], is based on the so-called extended Dupuit–Forch–heimer assumption,
that is,

(A3) the groundwater flow is parallel to the inclined impermeable bed and the piezometric head
is constant in the normal direction to the inclined impermeable bed.

The latter approach was used, e.g., in [6, 61, 63].
Very interesting discussions about the validity, use, and comparison of these two approaches

can be found, e.g., in [66] and introductions of papers [6, 8]. In Towner [61], the second approach
(flow parallel to inclined impermeable bed) was used to study groundwater flow between parallel
ditches in case of uniform rainfall. It was found that the calculated water table heights are in
much better agreement with experimental data than those published in [49] and calculated using
the first approach assuming horizontal flow. Based on this, we adopt the second approach in the
model developed in this paper.

2.5. Physical assumptions of our model. We assume that there is an inclined layer of imper-
meable bed (such as, e.g., bedrock or impermeable geosynthetic material) covered by a parallel
layer of permeable material (such as, e.g., permeable rock, soil, sand, gravel or some geosynthetic
porous material). The groundwater moves within the permeable layer in the saturated zone only
(see, e.g., [9, Sec. 1.1.2, p. 2 ]). We assume that the groundwater has a free surface, capillarity
effects above the free surface are neglected, and that the extended Dupuit-Forchheimer assump-
tion (A3) for the free surface is valid, i.e., the piezometric head h( · , t) is constant in the direction
perpendicular to the impermeable bed at any given time t. For simplicity, we further assume that
there is no bulk flow in the y direction. Thus the piezometric head is a function of x and t only
and given (x, t), the free surface is at the distance h(x, t) from the impermeable bed, see Figure 1.
We assume that the groundwater flow through porous medium forming the permeable layer is
governed by the power law (2.2) which in our situation is reduced to

v(x, t) = −c
∣∣∂h
∂x

(x, t)
∣∣m−1 ∂h

∂x
(x, t) .

Here, v(x, t) is flow velocity, and h(x, t) is piezometric head measured from the chosen horizontal
reference level, see Figure 1. Then, the flux per unit width in the saturated zone (denoted by



6 P. GIRG, L. KOTRLA EJDE-2025/57

φ

h(−1)

h(1)

ĥ

h

ĥ cosφ

x sinφ

x

z

A

B

C

x = −1

x = 1

????

rain

Figure 2. Geometric relations between h, ĥ and φ. Due to (A3) Piezometric
head h is assumed to be constant along the line segment AB. The fictive piezome-
ter is measuring piezometric head at point C.

Qp.u.w.) at any position x (see Figure 1) and any time t is

Qp.u.w.(x, t) = ĥ(x, t)v(x, t) = −cĥ(x, t)
∣∣∂h
∂x

(x, t)
∣∣m−1 ∂h

∂x
(x, t) . (2.3)

Here, ĥ(x, t) is thickness of the saturated zone measured perpendicularly to the inclined imperme-
able bed, see Figure 1. By the extended Dupuit-Forchheimer assumption (A3), the piezometric
head h(x, t) is constant along planes perpendicular to impermeable bed. Thus

h(x, t) = ĥ(x, t) cosφ+ x sinφ , (2.4)

where φ is angle of inclination, see Figure 1.
The equation of continuity for our geometric configuration reads as follows

∂ĥ

∂t
(x, t) +

∂Qp.u.w.

∂x
(x, t) = r(x, t) cosφ , (2.5)

since ĥ(x, t) is measured in perpendicular direction to the bedrock and thus represents the amount
of groundwater in the saturated zone per length x and unit width. The source function r (rain)
is given in the vertical direction. Thus r(x, t) has to be multiplied by cosφ. Combining equations
(2.3), (2.4) and (2.5), we obtain

∂ĥ

∂t
(x, t)

− c
∂

∂x

[
ĥ(x, t)

∣∣∣∂ĥ
∂x

(x, t) cosφ+ sinφ
∣∣∣
m−1(∂ĥ

∂x
(x, t) cosφ+ sinφ

)]

= r(x, t) cosφ .

(2.6)



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 7

unit

width free surface

impermeable layer

(a) the 3D geometric configuration.

φ

x sinφ

ĥ(x, t)

x

v

A

Bh(x, t)

free surface

impermeable layer

(b) 2D longitudinal-vertical section with state and flow variables marked.

Figure 3. State and flow variables in the case of the flow in porous medium
layered over inclined impermeable layer under generalized Dupuit-Forchheimer
assumption (A3). The averaged velocity is assumed to be constant on the line

segment AB and parallel to the inclined bedrock. Here, h(x, t) = ĥ(x, t) cosφ +
x sinφ, v = v(x, t) = |∂h/∂x|m−1∂h/∂x, and the groundwater flow per unit width

is Qp.u.w.(x, t) = ĥ(x, t)v(x, t).

2.6. Initial-boundary value problem for a mathematical model of groundwater flow

between infinite parallel ditches. From now on, we will use the parameter p
def
= m + 1 > 1

instead of m > 0, in order to follow standard notation used by “p-Laplacian community”, whereas
m is mostly used in hydrological literature. As usual, p′ > 1 denotes the conjugate exponent to
p > 1, i.e., 1/p+ 1/p′ = 1, in the sequel of the paper.

Let the constants H−1, H1 ≥ 0 denote the water level measured from the impermeable bed
in the left (x = −1) and right (x = 1) ditch, respectively. Let the function h0(x), x ∈ (−1, 1),
describes the initial state of the water level between the ditches. The water level is measured
perpendicularly to the impermeable bed here (see Section 2.4 for more details). To simplify



8 P. GIRG, L. KOTRLA EJDE-2025/57

notation, we also define f(x, t)
def
= r(x, t) cosφ. Then, based on (2.6), we obtain initial boundary

value problem

∂ĥ

∂t
(x, t)− c

∂

∂x

[
ĥ(x, t)

∣∣∣∂ĥ
∂x

(x, t) cosφ+ sinφ
∣∣∣
p−2(∂ĥ

∂x
(x, t) cosφ+ sinφ

)]

= f(x, t) , x ∈ (−1, 1), t > 0 ,

ĥ(−1, t) = H−1 and ĥ(1, t) = H1 , t > 0 ,

ĥ(x, 0) = h0(x) , x ∈ (−1, 1) .

(2.7)

Existence, weak comparison principles, and asymptotic behavior of bounded weak solutions for
this type of initial-boundary value problems have been studied in [21, 22]. Note that the analysis
(though very thorough) presented in that two aforementioned papers, is not complete yet and
there are many interesting and difficult open problems waiting for their resolution. It turns out
from the results and thorough discussions in [21, 22], that the properties of the steady states play
important role in the theory for the bounded weak solutions to the boundary-initial value problem
(2.7). This is one of the motivations for our detailed study of the steady-states of (2.7) presented
in this paper.

For the sake of keeping the amount of technicalities to the minimum, we limit ourselves to

the case of the same water level in both ditches, i.e., H
def
= H−1 = H1 > 0. Note that we also

assume H > 0 in this paper, since the case H = 0 is essentially different and much more difficult

to analyze. Let us observe that ĥ(x, t) ≡ const. = H is the steady state of (2.7) for f ≡ 0, and

ĥ(−1, t) = ĥ(1, t) ≡ const. = H. Our aim is to study perturbations of this steady-state solution
caused by rain or evaporation constant in time, which is incorporated into the model by considering

f ̸≡ 0. For this reason, we will write the steady-state solution in the form ĥ(x, t) = u(x) +H and
rewrite the equation for steady states of (2.7) in terms of unknown function u:

− d

dx

[
(u(x) +H)

∣∣du
dx

(x) cosφ+ sinφ
∣∣p−2

(du
dx

(x) cosφ+ sinφ
)]

= f(x) , x ∈ (−1, 1) ,

u(−1) = u(1) = 0 ,

(2.8)

where in the case of steady-states we must assume that the function f is independent of the
time variable and thus we write f(x) instead of f(x, t). We will use this notation from now on.
Moreover, it is natural to assume that the water level cannot drop below the impermeable layer
and thus we will consider only solutions that satisfy u(x) ≥ −H for x ∈ (−1, 1).

We define weak solution to the problem (2.8) as follows.

Definition 2.1 (Weak solution). Let p > 1, 0 < φ < π/2, and f ∈ L1(−1, 1) be given. By a weak

solution to (2.8) we mean a function u ∈ W 1,p
0 (−1, 1), u ≥ −H, satisfying

∫ 1

−1

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) v′(x)dx

=

∫ 1

−1

f(x)v(x)dx

(2.9)

for all v ∈ W 1,p
0 (−1, 1), where u′ and v′ stands for the respective weak derivatives.

Remark 2.2. Let us note that any u ∈ W 1,p
0 (−1, 1) has a representative in AC[−1, 1] (space of

absolutely continous functions on [−1, 1]). Thus every weak solution in our sense is also a bounded
weak solution. So our definition is compatible with results from [21, 22].

Moreover, let us note that the classical derivative of the AC[−1, 1] representative of u ∈
W 1,p(−1, 1) exists a.e. in [−1, 1], belongs to Lp(−1, 1), and coincides with the weak derivative
of u a.e. in [−1, 1] (see, e.g., [70, Thm. 2.1.4]). Therefore, we will use the same notation u′

for both the weak and classical derivatives of u with respect to x. Also, when we speak about
higher regularity of weak solutions such as smoothness, we mean that this regularity refers to the
AC[−1, 1] representative of u ∈ W 1,p(−1, 1).



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 9

Finally, let us note that our definition of weak solution is not the most general one. Indeed, it
is enough to assume that

(u(·) +H) |u′(·) cosφ+ sinφ|p−2
(u′(·) cosφ+ sinφ) ∈ Lp′

(−1, 1) (2.10)

in order that the integral on the left hand side of (2.9) makes sense. Indeed, a function u ̸∈
W 1,p

0 (−1, 1) satisfying (2.10) is, e.g., u(x) = |x|α − 1, if H = 1 and

0 <
(p− 1

p

)2

< α ≤ p− 1

p
< 1 .

However, dealing with such more general weak solutions is quite delicate. Especially, choice of
proper function spaces and finding additional conditions to obtain physically relevant solutions
are difficult questions. For that reason, we limit ourselves to weak solutions as defined above in
this paper.

3. Properties of weak solution

3.1. Regularity results. In this section we will show that any weak solution u to (2.9) satisfies
u > −H and u ∈ C1[−1, 1], provided the following hypothesis on f is imposed.

(A4) Assume that f ∈ L1(−1, 1) and satisfies

min
x0∈[−1,1]

min
x∈[x0,1]

(
H

p
p−1 +

p

(p− 1) cosφ

∫ 1

x

Φp′

(∫ τ

x0

f(s)ds
)
dτ

)
> 0 ,

where Φq(z)
def
= |z|q−1 sign z, for q > 1.

Remark 3.1. Let us note that Hypothesis (A4) is satisfied for any H > 0 and f ≥ 0 a.e.
in (−1, 1). This will turn out to be very usefull in our results concerning maximum principles
(Proposition 3.19 and Theorem 3.24).

Our proofs in this section rely on rewriting the weak formulation (2.9) as a differential equation
of the first order.
From the weak formulation to a differential equation of the first order. For any test function
v ∈ C∞

c (−1, 1), we can integrate the right-hand side of weak formulation (2.9) by parts and
obtain

∫ 1

−1

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) v′(x)dx

= −
[
v(x)

∫ 1

x

f(σ)dσ
]1
−1

+

∫ 1

−1

v′(x)

∫ 1

x

f(σ)dσdx

which yields
∫ 1

−1

[
(u(x) +H) |u′(x) cosφ+ sinφ|p−2

(u′(x) cosφ+ sinφ)−
∫ 1

x

f(σ)dσ
]
v′(x)dx = 0 (3.1)

for all v ∈ C∞
c (−1, 1). This means that the distributional derivative of the expression “[. . . ]”

in (3.1) is zero. Taking into account that u ∈ W 1,p
0 (−1, 1) ↪→ AC[−1, 1], the expression “[. . . ]”

belongs to L1(−1, 1). Thus by [38, Lem. 1.2.1., p. 13], we obtain

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ)−

∫ 1

x

f(σ)dσ

= κ ≡ const. a. e. in [−1, 1] .

(3.2)

Theorem 3.2. Let u be a weak solution to (2.8) and x1 ∈ [−1, 1) be such that

u(x) > −H for all [x1, 1] . (3.3)

Then u|[x1,1] ∈ C1[x1, 1].



10 P. GIRG, L. KOTRLA EJDE-2025/57

Proof. The assumption (3.3) allows us to rewrite (3.2) as

|u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ)

=
1

(u(x) +H)

(
κ+

∫ 1

x

f(σ)dσ
)

a.e. in [x1, 1] .
(3.4)

Then we obtain

u′(x) =
1

cosφ
Φp′

( 1

(u(x) +H)

(
κ+

∫ 1

x

f(σ)dσ
))

− tanφ a.e. in [x1, 1] . (3.5)

In the remainder of this proof, u will be identified with the AC[−1, 1] representative of u ∈
W 1,p

0 (−1, 1). Now let us define function F : R× [x1, 1] → R,

F (z, x)
def
= z − 1

cosφ
Φp′

( 1

(u(x) +H)

(
κ+

∫ 1

x

f(σ)dσ
))

+ tanφ .

It follows from u ∈ AC[−1, 1] and (3.3) that the function F is continuous on R× [x1, 1]. Moreover,
it is strictly increasing in the first variable and

lim
z→±∞

F (z, x) = ±∞ for all x ∈ [x1, 1] .

Thus for each x ∈ [x1, 1] there is unique z(x) such that

F (z(x), x) = 0 . (3.6)

Indeed,

z(x) =
1

cosφ
Φp′

( 1

(u(x) +H)

(
κ+

∫ 1

x

f(σ)dσ
))

+ tanφ ,

which is continuous on [x1, 1] since u ∈ AC[−1, 1] and (3.3) holds. From (3.5), we also know that
F (u′(x), x) = 0 a.e. in [x1, 1]. Since z(x) is the only solution to (3.6) for any given x ∈ [x1, 1], it
must hold

z(x) = u′(x) a.e. in [x1, 1] .

Now, as u ∈ AC[−1, 1], we have u(x) =
∫ x

x1
u′(σ)dσ =

∫ x

x1
z(σ) dσ for all x ∈ [x1, 1]. Taking into

account that z ∈ C[x1, 1], we see that the classical derivative of u exists in every point in x ∈ (x1, x)
and it is continuously extendable to [x1, 1]. Hence the restriction to [x1, 1] of u ∈ AC[−1, 1] belongs
to C1[x1, 1]. □

Corollary 3.3. Let u be a weak solution to (2.8) and x1 is as in Theorem 3.2. Then

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ)

−H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

=

∫ 1

x

f(σ)dσ

(3.7)

holds pointwise everywhere in [x1, 1].

Proof. Since u|[x1,1] ∈ C1[x1, 1] by Theorem 3.2, we may evaluate (3.2) at x = 1 to obtain

κ = (u(1) +H) |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)−

∫ 1

1

f(σ)dσ

= H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ) .

(3.8)

Combining (3.2) and (3.8), we obtain (3.7) which holds pointwise everywhere in [x1, 1]. □

Now, assuming Hypothesis (A4), we will show that u > −H on the whole [−1, 1] and, conse-
quently, u ∈ C1[−1, 1].



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 11

Lemma 3.4. Let x0 ∈ (−1, 1) and u be a weak solution to (2.8) such that u(x0) = −H and
u(x) > −H for all x ∈ (x0, 1). Then

H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ) = −

∫ 1

x0

f(x)dx . (3.9)

Proof. Let us use

vε(x)
def
=





0 for x ∈ (−1, x0] ,

(x− x0)/ε for x ∈ (x0, x0 + ε) ,

1 for x ∈ [x0 + ε, 1− ε] ,

(1− x)/ε for x ∈ (1− ε, 1)

as a test function in (2.9), with 0 < ε < (1− x0)/2. Then

∫ 1

−1

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) v′ε(x)dx =

∫ 1

−1

f(x)vε(x)dx

becomes

1

ε

∫ x0+ε

x0

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) dx

− 1

ε

∫ 1

1−ε

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) dx

=

∫ 1

−1

f(x)vε(x)dx .

(3.10)

Using that u ∈ C1[x0 + δ, 1] for any 0 < δ < 1− x0 by Theorem 3.2, we find that

lim
ε→0+

1

ε

∫ 1

1−ε

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) dx

= H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

(3.11)

since x = 1 is a Lebesgue point of the integrand and u(1) = 0. By the Lebesgue dominated
convergence theorem, we also find that

lim
ε→0+

∫ 1

−1

f(x)vε(x)dx =

∫ 1

x0

f(x)dx .

Thus

H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

= −
∫ 1

x0

f(x)dx+ lim
ε→0+

1

ε

∫ x0+ε

x0

(u(x) +H)
∣∣u′(x) cosφ

+ sinφ
∣∣p−2

(u′(x) cosφ+ sinφ) dx .

(3.12)

We will prove (3.9) by showing that the last term vanishes. For each x ∈ (x0, x0 + ε), we have

|u(x) +H|
ε1/p′ =

|u(x)− (−H)|
ε1/p′ =

∣∣ ∫ x

x0
u′(s)ds

∣∣
ε1/p′

≤
∫ x

x0
|u′(s)|ds

(x− x0)1/p
′ ≤

( ∫ x

x0
1p

′
ds

)1/p′

(x− x0)1/p
′

(∫ x

x0

|u′(s)|pds
)1/p

≤
(∫ x0+ε

x0

|u′(s)|pds
)1/p

.

(3.13)



12 P. GIRG, L. KOTRLA EJDE-2025/57

On the other hand,
∫ x0+ε

x0
|u′(x) cosφ+ sinφ|p−1dx

ε1/p
≤

( ∫ x0+ε

x0
1pdx

)1/p

ε1/p

(∫ x0+ε

x0

|u′(x) cosφ+ sinφ|pdx
)1/p′

=
(∫ x0+ε

x0

|u′(x) cosφ+ sinφ|pdx
)1/p′

.

(3.14)

Now, combining (3.13) and (3.14), we obtain

lim
ε→0+

1

ε

∣∣∣
∫ x0+ε

x0

(u(x) +H)|u′(x) cosφ+ sinφ|p−2 (u′(x) cosφ+ sinφ) dx
∣∣∣

≤ lim
ε→0+

1

ε1/p

∫ x0+ε

x0

|u(x) +H|
ε1/p′ |u′(x) cosφ+ sinφ|p−1

dx

≤ lim
ε→0+

1

ε1/p

∫ x0+ε

x0

(∫ x0+ε

x0

|u′(x)|pdx
)1/p

|u′(x) cosφ+ sinφ|p−1
dx

≤ lim
ε→0+

(∫ x0+ε

x0

|u′(x)|pdx
)1/p(∫ x0+ε

x0

|u′(x) cosφ+ sinφ|pdx
)1/p′

= 0 ,

(3.15)

since u ∈ W 1,p
0 (−1, 1). Thus the last term in (3.12) vanishes and the proof is complete. □

Theorem 3.5. Let Hypothesis (A4) hold. Then any weak solution u to (2.8) satisfies u > −H on
[−1, 1].

Proof. Assume by contradiction that there is at least one ξ ∈ (−1, 1) such that u(ξ) = −H. Since

u ∈ W 1,p
0 (−1, 1) ↪→ AC[−1, 1] and u(1) = 0, we may choose ξ = x0 such that u(x) > −H for all

x ∈ (x0, 1). Then by Corollary 3.3, we have

H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

− (u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ)

= −
∫ 1

x

f(σ)dσ

(3.16)

for all x ∈ [x0 + δ, 1) with any 0 < δ < 1− x0. Using (3.9), we find

− (u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) = −

∫ 1

x

f(σ)dσ +

∫ 1

x0

f(σ)dσ

=

∫ x

x0

f(σ)dσ

for all x ∈ (x0, 1). Using the substitution w(x) = u(x) +H and denoting F̃ (x) =
∫ x

x0
f(s)ds, we

obtain
w(x)Φp (w

′(x) cosφ+ sinφ) = −F̃ (x)

and, equivalently,

Φp

(
[w(x)]

1
p−1w′(x) cosφ+ [w(x)]

1
p−1 sinφ

)
= −F̃ (x) .

Since Φp′ is an inverse to Φp, we obtain

[w(x)]
1

p−1w′(x) cosφ+ [w(x)]
1

p−1 sinφ = −Φp′
(
F̃ (x)

)

Using the substitution v(x) = [w(x)]
p

p−1 , we have the ODE

p− 1

p
v′(x) cosφ+ v1/p(x) sinφ = −Φp′

(
F̃ (x)

)
,

v(1) = H
p

p−1 ,

v(x) ≥ 0 ,

φ > 0 .



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 13

Since v(x) ≥ 0, we have
p− 1

p
v′(x) cosφ ≤ −Φp′

(
F̃ (x)

)

and integrating from x to 1 we obtain

(p− 1) cosφ

p
(v(1)− v(x)) ≤ −

∫ 1

x

Φp′
(
F̃ (τ)

)
dτ .

It follows that

v(x) ≥ H
p

p−1 +
p

(p− 1) cosφ

∫ 1

x

Φp′
(
F̃ (τ)

)
dτ .

By (A4), we have

K = min
x0∈[−1,1]

min
x∈[x0,1]

(
H

p
p−1 +

p

(p− 1) cosφ

∫ 1

x

Φp′

(∫ τ

x0

f(s)ds
)
dτ

)
> 0 ,

then v(x) ≥ K must hold for all x ∈ (x0, 1). This contradicts v(x0) = 0 as v is continuous on
[−1, 1]. □

Theorem 3.6 (C1-regularity of a weak solution). Let Hypothesis (A4) hold. Then any weak
solution to (2.8) satisfies u ∈ C1[−1, 1].

Proof. By Theorem 3.5, u(x) > −H for all x ∈ [−1, 1]. Then we may choose x1 = −1 in
Theorem 3.2 to obtain that u ∈ C1[−1, 1]. □

Corollary 3.7. Under Hypothesis (A4), the equality

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ)

−H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

=

∫ 1

x

f(σ)dσ

(3.17)

holds pointwise for all x ∈ [−1, 1].

Proof. The proof of (3.17) is similar as the proof of Corollary 3.3 with x1 = −1 since u ∈ C1[−1, 1]
by Theorem 3.6. □

3.2. A priori bound on a weak solution. Here, we use the regularity result from the previous
section to obtain a priori bound on the L∞-norm of a weak solution to (2.9) depending on φ and
∥f∥L1(−1,1). The proof relies on the first order formula (3.17).

Theorem 3.8. Assume that Hypothesis (A4) holds. Then every weak solution u to (2.8) satisfies

∥u∥∞ ≤ ∥f∥L1(−1,1)

(sinφ)p−1
. (3.18)

Proof. We distinguish four basic cases further possibly divided into subcases.

Case 1. u ≡ 0. Statement is satisfied trivially.

Case 2: u(x) ≥ 0, u(xmax) > 0. Maximum is achieved in the interior point xmax ∈ (−1, 1). Thus
u′(xmax) = 0 since u ∈ C1[−1, 1]. Minimum is achieved at the boundary. Evaluating (3.17) at
xmax, we obtain

(u(xmax) +H) (sinφ)
p−1 −H |u′(1) cosφ+ sinφ|p−2

(u′(1) cosφ+ sinφ)

=

∫ 1

xmax

f(σ)dσ .
(3.19)

Since u ∈ C1[−1, 1], u(−1) = 0 = u(1), and u(x) ≥ 0, we have

u′(−1) ≥ 0 ≥ u′(1) .

Subcase 2a: u′(1) < − tanφ. Then u′(1) cosφ+ sinφ < 0. Hence

−H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ) = H |u′(1) cosφ+ sinφ|p−1

> 0 .



14 P. GIRG, L. KOTRLA EJDE-2025/57

Then it follows from (3.19) that

0 < u(xmax) ≤ u(xmax) +H +
H

(sinφ)p−1
|u′(1) cosφ+ sinφ|p−1

=
1

(sinφ)p−1

∫ 1

xmax

f(σ)dσ

≤ ∥f∥L1(−1,1)

(sinφ)p−1
.

Subcase 2b: 0 ≥ u′(1) ≥ − tanφ. Then 0 ≤ u′(1) cosφ + sinφ ≤ sinφ. Thus it follows from
(3.19) that

0 < u(xmax) ≤ u(xmax) +H
(
1−

( |u′(1) cosφ+ sinφ|
sinφ

)p−1)

=
1

(sinφ)p−1

∫ 1

xmax

f(σ)dσ

≤ ∥f∥L1(−1,1)

(sinφ)p−1
.

Case 3: u(x) ≤ 0, u(xmin) < 0. Minimum is achieved in the interior point xmin ∈ (−1, 1). Thus
u′(xmin) = 0 since u ∈ C1[−1, 1]. Maximum is achieved at the boundary. From u ∈ C1[−1, 1],
u(−1) = 0 = u(1), and u(x) ≤ 0, we conclude that

u′(−1) ≤ 0 ≤ u′(1) .

Evaluating (3.17) at xmin, we obtain

(u(xmin) +H) (sinφ)
p−1 −H |u′(1) cosφ+ sinφ|p−2

(u′(1) cosφ+ sinφ) =

∫ 1

xmin

f(σ)dσ . (3.20)

Since u(xmin) < 0 and u′(1) > 0, we have

(|u(xmin)| −H) (sinφ)
p−1

+H |u′(1) cosφ+ sinφ|p−1
= −

∫ 1

xmin

f(σ)dσ ,

from which it follows that

0 < |u(xmin)| ≤ |u(xmin)|+H
(( |u′(1) cosφ+ sinφ|

sinφ

)p−1

− 1
)

= − 1

(sinφ)p−1

∫ 1

xmax

f(σ)dσ ≤ ∥f∥L1(−1,1)

(sinφ)p−1
.

Case 4: u(xmin) < 0 < u(xmax). In this case, xmin ̸= xmax and xmin, xmax ∈ (−1, 1). Without
loss of generality assume that xmin < xmax. Since u ∈ C1[−1, 1], u′(xmin) = 0 = u′(xmax). Hence
both (3.19) and (3.20) are valid. Subtracting (3.20) from (3.19), we obtain

0 < (u(xmax)− u(xmin)) (sinφ)
p−1 ≤ −

∫ xmax

xmin

f(σ)dσ ≤ ∥f∥L1(−1,1) .

Since u(xmin) < 0, we obtain

0 < u(xmax) ≤ u(xmax)− u(xmin) ≤
∥f∥L1(−1,1)

(sinφ)p−1
.

Moreover, by u(xmax) > 0, we also have

0 < |u(xmin)| < u(xmax)− u(xmin) ≤
∥f∥L1(−1,1)

(sinφ)p−1
.

Finally, we find that

0 ≤ ∥u∥∞ ≤ ∥f∥L1(−1,1)

(sinφ)p−1



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 15

since these four are the all possible cases. □

3.3. Existence of a weak solution. Now we establish an existence result for the problem (2.8)
using combination of a priori estimate (3.18) and classical theory of pseudomonotone operators.
Since this theory is well known, we refer the reader to, e.g., [46, p. 5; Def. 2.1, pp. 31–32; and
Def. 2.5, p. 33] for definitions of bounded, coercive, and pseudomonotone operator, respectively.

As usual, dual of W 1,p
0 (−1, 1) will be denoted by W−1,p′

(−1, 1). Let us note that a priori estimate
(3.18) plays a key role in verification that the nonlinear term in (2.8) satisfies structural conditions
(presented in [46]), which allow application of the theory of pseudomonotone operators. Our
existence result reads as follows.

Theorem 3.9 (Existence of a weak solution). Assume that

∥f∥L1(−1,1) < H(sinφ)p−1 . (3.21)

Then the problem (2.8) possesses at least one weak solution.

Proof. Set

k
def
=

∥f∥L1(−1,1)

(sinφ)p−1

and define truncation function

Tk(r)
def
=





−k, r < −k

r, −k ≤ r ≤ k ,

k, r > k .

We say that w ∈ W 1,p
0 (−1, 1) satisfies truncated version of (2.9) if

∫ 1

−1

(Tk(w(x)) +H) |w′(x) cosφ+ sinφ|p−2
(w′(x) cosφ+ sinφ) v′(x)dx

=

∫ 1

−1

f(x)v(x)dx

(3.22)

for all v ∈ W 1,p
0 (−1, 1). Now we briefly comment on regularity properties and a priori bounds

valid for solutions to (3.22), omitting details since the proofs are analogous to and even simpler
than for the case of solutions to (2.9). Considering (3.21), we find that Tk(w(x)) > −H for all
x ∈ [−1, 1]. This ensures that the argument used in the proof of Theorem 3.2 can be applied
directly to the entire interval [−1, 1]. Consequently, we obtain w ∈ C1[−1, 1]. An analogous result
to Corollary 3.7 holds for solutions to (3.22), without assuming Hypothesis (A4). Specifically, any
solution w to (3.22) satisfies

(Tk(w(x)) +H) |w′(x) cosφ+ sinφ|p−2
(w′(x) cosφ+ sinφ)

−H |w′(1) cosφ+ sinφ|p−2
(w′(1) cosφ+ sinφ)

=

∫ 1

x

f(σ)dσ

(3.23)

holds pointwise for all x ∈ [−1, 1]. Finaly, the a priori estimate

∥w∥∞ ≤ ∥f∥L1(−1,1)

(sinφ)p−1
(3.24)

follows from (3.23) using the same steps as in the proof of Theorem 3.8. Now, in view of (3.24),
(3.21), and the definition of k, we see that any solution to (3.22) also satisfies (2.9).

We now proceed to establish the existence of solution to (3.22), which will then imply the
existence of solution to (2.9), which is a weak solutions to (2.8). This will be achieved by verifying,

as detailed in Appendix 5, that the operator A : W 1,p
0 (−1, 1) → W−1,p′

(−1, 1) defined by the left-
hand side of (3.22) is a bounded, coercive, and pseudomonotone operator. By [46, Thm. 2.6, p. 33],

there exists a solution w ∈ W 1,p
0 (−1, 1) to the operator equation A(w) = g for all g ∈ W−1,p′

(−1, 1)



16 P. GIRG, L. KOTRLA EJDE-2025/57

and hence also for g ∈ W−1,p′
(−1, 1) defined by the right-hand side of (3.22), which concludes the

proof. □

Remark 3.10. In Theorem 3.9, we obtained solutions to (2.9) using the fact that any solution
to (3.22) is also a solution to (2.9) under the condition (3.21). This raises the question of whether
(2.9) might possess solutions that do not also satisfy (3.22). While we can only provide partial
answer to this question, we can establish the following: If we assume Hypothesis (A4) in addition
to (3.21), then any weak solution to (2.9) must also satisfy (3.22). Consequently, the two problems
become equivalent under this combined assumption.

3.4. Linearization at the trivial solution. The method of linearization of the p-Laplacian
at some given solution has been succesfuly used in treating various questions such as validity of
comparison principles for elliptic and parabolic problems involving p-Laplacian, see, e.g., [11, 17,
18, 32], Fredholm Alternative for the p-Laplacian [23, 60]. The reader who is further interested in
this method is refered to [60] for the most detailed description of this method.

The main tool of this method is the zero order Taylor formula for the power function with the
remainder in the integral form:

|a|p−2a− |b|p−2b = (p− 1)

∫ 1

0

|b+ θ(a− b)|p−2 dθ (a− b) (3.25)

To estimate the remainder term from below (the more involved estimate), we use the following
lemma, which is an alternative to [60, Lem. A.1, p. 233] for one-dimensional case. Main advantage
of our approach is that it is better suited for a particular form of reminder term in calculations
below and that it provides simple explicit lower bound.

Lemma 3.11. Let p > 2 and a ∈ R, then

(p− 1)

∫ 1

0

|1 + θa|p−2 dθ ≥ 1/2

Proof. We distinguish two cases.

Case 1. Let a ≥ −2. Then 1 + θa ≥ 1− 2θ ≥ 0 for θ ∈ [0, 1/2]. Thus

(p− 1)

∫ 1

0

|1 + θa|p−2
dθ ≥ (p− 1)

∫ 1/2

0

|1 + θa|p−2
dθ

≥ (p− 1)

∫ 1/2

0

|1− 2θ|p−2
dθ = 1/2 .

Case 2. Let a < −2. Then 1 + θa ≤ 1− 2θ ≤ 0 for θ ∈ [1/2, 1] and we again obtain

(p− 1)

∫ 1

0

|1 + θa|p−2
dθ ≥ (p− 1)

∫ 1

1/2

|1 + θa|p−2
dθ

≥ (p− 1)

∫ 1

1/2

|1− 2θ|p−2
dθ = 1/2 .

This completes the proof. □

Now we are ready to state our main result concerning linearization of (2.8) at the zero solution.

Lemma 3.12. Let p > 2 and Hypothesis (A4) be satisfied. Let u be a weak solution to nonlinear
problem (2.8) and for such u let us define

D(x)
def
= (u(x) +H)(p− 1)

∫ 1

0

|sinφ+ θu′(x) cosφ|p−2
dθ cosφ (3.26)

for every x ∈ [−1, 1]. Then D(·) ∈ C[−1, 1] and, for all x ∈ [−1, 1], the following estimate holds

D(x) ≥ K
(
φ,H, p, ∥u−∥∞

)
> 0 . (3.27)



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 17

Moreover, u is also the weak solution to the linear problem

− d

dx

(
D(x)

du

dx
(x)

)
− (sinφ)p−1 du

dx
(x) = f(x)

u(−1) = 0 = u(1) ,
(3.28)

that is, u satisfies
∫ 1

−1

D(x)u′(x)v′(x) dx+ (sinφ)p−1

∫ 1

−1

u(x)v′(x) dx =

∫ 1

−1

f(x)v(x)dx (3.29)

for all v ∈ W 1,p
0 (−1, 1).

Proof. Let us consider any (but fixed) weak solution u to (2.8). Since u ∈ C1[−1, 1] by Theorem 3.6
and u(x) > −H, by Theorem 3.5, there exists M > 0 depending on ∥u−∥∞ such that

min
x∈[−1,1]

(u(x) +H) ≥ M > 0 .

Then the function D(·) given by (3.26) is well defined on [−1, 1] and D(·) ∈ C[−1, 1] for p > 2.
Moreover,

D(x) = (u(x) +H)(p− 1)

∫ 1

0

|sinφ+ θu′(x) cosφ|p−2
dθ cosφ

≥ M(sinφ)p−2(p− 1)

∫ 1

0

|1 + θu′(x) cotφ|p−2
dθ cosφ

≥ M

2
(sinφ)p−2 cosφ > 0 ,

(3.30)

where we used Lemma 3.11 to estimate (p−1)
∫ 1

0
. . . dθ ≥ 1/2. Thus by settingK(φ,H, p, ∥u−∥∞) =

M
2 (sinφ)p−2 cosφ, we established validity of (3.27).
Again, let us consider any (but fixed) weak solution u to (2.8). Let the function D(·) be

constructed from this fixed u by (3.26). It remains to show that u satisfy (3.29). To do this, let
us observe that u ≡ 0 satisfies (2.9) for f ≡ 0. Indeed, we have

∫ 1

−1

H| sinφ|p−2 sinφ · v′(x) dx =

∫ 1

−1

0 · v(x) dx (3.31)

for each v ∈ W 1,p
0 (−1, 1). Now subtracting (3.31) from (2.9), i.e., from
∫ 1

−1

(u(x) +H) |u′(x) cosφ+ sinφ|p−2
(u′(x) cosφ+ sinφ) v′(x) dx

=

∫ 1

−1

f(x)v(x) dx ,

we obtain
∫ 1

−1

[
(u(x) +H) |u′(x) cosφ+ sinφ|p−2

(u′(x) cosφ+ sinφ)−H(sinφ)p−1
]
v′(x) dx

=

∫ 1

−1

f(x)v(x) dx .

(3.32)

By “adding zero” 0 = u(x)(sinφ)p−1 − u(x)(sinφ)p−1 to the term “[. . . ]” above, we obtain

[. . . ] = (u(x) +H)︸ ︷︷ ︸
⩾const.>0

[
|u′(x) cosφ+ sinφ|p−2

(u′(x) cosφ+ sinφ)− | sinφ|p−2 sinφ
]

︸ ︷︷ ︸
(p−1)

∫ 1
0
|sinφ+θu′(x) cosφ|p−2dθ u′(x) cosφ

+ u(x)(sinφ)p−1

= (u(x) +H)(p− 1)

∫ 1

0

|sinφ+ θu′(x) cosφ|p−2
dθ u′(x) cosφ+ u(x)(sinφ)p−1

= D(x)u′(x) + u(x)(sinφ)p−1 .



18 P. GIRG, L. KOTRLA EJDE-2025/57

Using the last formula on the last line above instead of [. . . ] in (3.32), we obtain
∫ 1

−1

D(x)u′(x)v′(x) dx+ (sinφ)p−1

∫ 1

−1

u(x)v′(x) dx =

∫ 1

−1

f(x)v(x)dx

for all v ∈ W 1,p
0 (−1, 1). This means that u is the weak solution to the linear problem (3.28). This

completes the proof. □

Let us note that, due to (3.26), the diffusion coefficient D(·) depends on a weak solution u
under consideration of the nonlinear problem (2.8). The following lemma provides lower bound
on the diffusion coefficient independent of a particular choice of a weak solution u to the nonlinear
problem (2.8).

Lemma 3.13. Let p > 2 and Hypothesis (A4) be satisfied. Moreover, let

∥f∥L1(−1,1) < H(sinφ)p−1 .

Then
D(x) ≥ K ′ (φ,H, p, ∥f∥L1(−1,1)

)

def
=

1

2

(
H − 1

(sinφ)p−1
∥f∥L1(−1,1)

)
(sinφ)p−2 cosφ > 0

(3.33)

for all x ∈ [−1, 1].

Proof. Taking into consideration (3.18), we can use

M =
(
H − 1

(sinφ)p−1
∥f∥L1(−1,1)

)

in (3.30). This completes the proof. □

Remark 3.14. Assume that f(x) < 0 for all x ∈ (−1, 1) and that
∫ 1

−1

f(x) dx = −H(sinφ)p−1 . (3.34)

Then

min
x0∈[−1,1]

min
x∈[x0,1]

(
H

p
p−1 +

p

(p− 1) cosφ

∫ 1

x

Φp′

(∫ τ

x0

f(s)ds
)
dτ

)

≥ H
p

p−1 − p

(p− 1) cosφ

∫ 1

−1

H
1

p−1 sinφdx

= H
p

p−1 − 2
p

(p− 1)
H

1
p−1 tanφ

= H
1

p−1

(
H − 2

p

(p− 1)
tanφ

)
.

Thus Hypothesis (A4) is satisfied for any H > 2 p
(p−1) tanφ. Hence, we obtain the lower bound

u > −H by Theorem 3.5. But the lower bound on D(·) obtained from this information would be
dependent on the concrete weak solution u of the nonlinear problem (2.9).

On the other hand, the bound (3.18), which is independent of u, is not optimal. Indeed, if a
negative function f satisfies (3.34), then

∥f∥L1(−1,1) = H(sinφ)p−1 ,

and by (3.18) from Theorem 3.8, u ≥ −H. Hence the lower bound of type (3.33) independent of u
obtained from (3.30), yields that D(x) ≥ const. ≥ 0 for all x ∈ [−1, 1], but in our further analysis
we need this constant to be strictly positive. Thus it will be very interesting for practical reasons
to obtain finer estimates of type (3.18) in order to get finer lower bounds of type (3.33). We leave
it as an interesting and technically quite complicated open problem.



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 19

Remark 3.15. Let us note that we can handle only the case p > 2, since we are lacking an
analogue of Lemma 3.11 for 1 < p < 2. We also do not know (at the time when we wrote this
paper), if the spatially dependent diffusion coefficient D(·) given by (3.26) is essentially bounded
for 1 < p < 2. If not, the weak solution of the linearized problem (3.28) has to be considered in
some appropriate weighted Sobolev spaces, see, e.g., [23, 24, 60]. Obtaining similar results such
as Lemma 3.11 and Lemma 3.12 for 1 < p < 2 poses an interesting and important open problem.

3.5. Additional regularity results. Our next goal is to find an estimate for ∥u′∥∞. Essential
tool of this section is the method of linearization presented in previous section. For this reason, we
can deal with the case p > 2 only. We will use the fact that any weak solution u to (2.8) satisfies
u > −H by Theorem 3.5 and hence for any weak solution u to (2.8) there exist M > 0 such that

min
x∈[−1,1]

(u(x) +H) ≥ M > 0 . (3.35)

The main result of this section is the following theorem.

Theorem 3.16. Let p > 2, Hypothesis (A4) be satisfied, and u be any (bounded) weak solution to
(2.8). If u′(1) ≥ 0, then

∥u′∥∞ ≤ 1

K (φ,H, p, ∥u−∥∞)

(
2 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) ,

else

∥u′∥∞ ≤ 2

K (φ,H, p, ∥u−∥∞)

(
1 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) ,

where K (φ,H, p, ∥u−∥∞) is the constant from Lemma 3.12, inequality (3.27).

Lemma 3.17. Let f ∈ L1(−1, 1) satisfy Hypothesis (A4) and u be any (bounded) weak solution
to (2.8). If u′(1) ≥ 0, then

u′(1) ≤ 1

H (sinφ)
p−2

cosφ
∥f∥L1(−1,1) ,

else

|u′(1)| ≤ 2

H cosφ (sinφ)
p−2 ∥f∥L1(−1,1) .

Proof. We distinguish three basic cases possibly divided into subcases. In all cases except the
first one we will use (3.17) evaluated at stationary point. Let us note that such point exists since

u ∈ W 1,p
0 (−1, 1) ↪→ C[−1, 1] with u(−1) = u(1) = 0, and u ∈ C1(−1, 1) by Theorem 3.6. Denote

S
def
= {x ∈ (−1, 1) : u′(x) = 0}. Then, for any x1 ∈ S, we obtain from (3.17)

H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

= (u (x1) +H) (sinφ)p−1 −
∫ 1

x1

f(σ) dσ .
(3.36)

Case 1: u′(1) = 0. Statement is satisfied trivially.

Case 2: u′(1) > 0. It follows that there exists x1 ∈ S satisfying u(x1) < 0. Then it follows from
(3.36) that

0 < H (sinφ)
p−2

(u′(1) cosφ+ sinφ)

≤ H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

= (u (x1) +H) (sinφ)p−1 −
∫ 1

x1

f(σ) dσ .

Subtracting H(sinφ)p−1 from the previous inequality we obtain

0 < H (sinφ)
p−2

u′(1) cosφ ≤ u (x1) (sinφ)
p−1 −

∫ 1

x1

f(σ) dσ .



20 P. GIRG, L. KOTRLA EJDE-2025/57

Using that u(x1) < 0, we finally obtain

0 < H (sinφ)
p−2

u′(1) cosφ

≤ u (x1) (sinφ)
p−1 −

∫ 1

x1

f(σ) dσ

<

∫ 1

x1

|f(σ)|dσ

≤ ∥f∥L1(−1,1) .

(3.37)

It follows

0 < u′(1) <
1

H (sinφ)
p−2

cosφ
∥f∥L1(−1,1) .

Case 3: u′(1) < 0. There exists x1 ∈ S satisfying u(x1) > 0. Further, this case is divided into
three subcases.
Subcase 3a: 0 > u′(1) ≥ − tanφ. Then 0 ≤ u′(1) cosφ+ sinφ < sinφ. Using (3.36) we obtain

H(sinφ)p−2(u′(1) cosφ+ sinφ)

≥ H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

= (u(x1) +H)(sinφ)p−1 −
∫ 1

x1

f(σ) dσ .

Hence

0 > H cosφ(sinφ)p−2u′(1) ≥ u(x1)(sinφ)
p−1 −

∫ 1

x1

f(σ) dσ > −
∫ 1

x1

f(σ) dσ .

It follows that

|u′(1)| < 1

H (sinφ)
p−2

cosφ
∥f∥L1(−1,1) .

Subcase 3b: − tanφ > u′(1) > −2 tanφ. Then

0 > u′(1) cosφ+ sinφ > − sinφ,

(sinφ)p−2 > |u(1) cosφ+ sinφ|p−2
.

Using (3.36) where we subtracted H(sinφ)p−1 from both sides and equality (3.25), we obtain

0 > H
(
|u′(1) cosφ+ sinφ|p−2

(u′(1) cosφ+ sinφ)− |sinφ|p−2
sinφ

)

= H(p− 1)

∫ 1

0

| sinφ+ θ(u′(1) cosφ+ sinφ)|p−2 dθu′(1) cosφ

= u(x1)(sinφ)
p−1 −

∫ 1

x1

f(σ) dσ .

Hence

0 < H(p− 1)

∫ 1

0

| sinφ+ θ(u′(1) cosφ+ sinφ)|p−2 dθ (−u′(1)) cosφ

= −u(x1)(sinφ)
p−1 +

∫ 1

x1

f(σ) dσ

<

∫ 1

x1

f(σ) dσ ,

from which it follows that

0 < |u′(1)| ≤ ∥f∥L1(−1,1)

H cosφ(p− 1)
∫ 1

0
| sinφ+ θ(u′(−1) cosφ+ sinφ)|p−2 dθ

.



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 21

Since

(p− 1)

∫ 1

0

|1 + θ(u′(1) cotφ+ 1)|p−2 dθ ≥ 1

2

by Lemma 3.11, we obtain an upper bound

0 < |u′(1)| ≤ 2
∥f∥L1(−1,1)

H cosφ(sinφ)p−2
.

Subcase 3c: −2 tanφ ≥ u′(1). Then

− sinφ ≥ u′(1) cosφ+ sinφ,

(sinφ)p−2 ≤ |u(1) cosφ+ sinφ|p−2
.

Using (3.36) we obtain

0 > H(sinφ)p−2(u′(1) cosφ+ sinφ)

≥ H |u′(1) cosφ+ sinφ|p−2
(u′(1) cosφ+ sinφ)

= (u(x1) +H)(sinφ)p−1 −
∫ 1

x1

f(σ) dσ .

The rest of the proof is similar to the one of Subcase 3a. □

Proof of Theorem 3.16. We will use linearization at the trivial solution described in Section 3.4.
Thus an arbitrary but fixed weak solution u to (2.8) also satisfies the following weak formulation
of the linear problem

∫ 1

−1

(
D(x)u′(x) + (sinφ)p−1u(x)−

∫ 1

x

f(σ)dσ
)
v′(x) dx = 0 (3.38)

with the diffusion coefficient given by (3.26) for the fixed weak solution u to (2.8). Hence

D(x)u′(x) + (sinφ)p−1u(x)−
∫ 1

x

f(σ) dσ = κ , (3.39)

where the constant

κ = D(1)u′(1) + (sinφ)p−1u(1) + cosφ

∫ 1

1

f(σ) dσ = D(1)u′(1)

by taking x = 1 in the previous equation. We substitute D(1)u′(1) for κ in (3.39) to obtain

D(x)u′(x) = −(sinφ)p−1u(x) +

∫ 1

x

f(σ) dσ +D(1)u′(1) .

Thus

|u′(x)| ≤ 1

minx∈[−1,1] D(x)

(
sin(φ)p−1|u(x)|+ ∥f∥L1(−1,1) +D(1)|u′(1)|

)
. (3.40)

by the triangle inequality. For u′(1) ≥ 0, we have

|u′(x)| ≤ 1

minx∈[−1,1] D(x)

(
(sinφ)p−1 1

(sinφ)p−1
+ 1 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) (3.41)

by Theorem 3.8 and Lemma 3.17. Using (3.27) we finally obtain

|u′(x)| ≤ 1

K (φ,H, p, ∥u−∥∞)

(
2 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) .

For u′(1) < 0, we have different bound on |u′(1)| and hence we obtain

|u′(x)| ≤ 1

minx∈[−1,1] D(x)

(
(sinφ)p−1 1

(sinφ)p−1
+ 1 +

2D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) (3.42)

by Theorem 3.8 and Lemma 3.17. Then we have

|u′(x)| ≤ 2

K (φ,H, p, ∥u−∥∞)

(
1 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1)



22 P. GIRG, L. KOTRLA EJDE-2025/57

using (3.27) again. □

The estimate on ∥u′∥∞ from the previous theorem can be further refined to become a priori
bound (independent of ∥u−∥∞ and of D(1)) under an additional condition.

Theorem 3.18. Let p > 2, Hypothesis (A4) be satisfied, and u be any (bounded) weak solution to
(2.8). Moreover, let there exists β > 0 such that

∥f∥L1(−1,1) ≤ β < H(sinφ)p−1 . (3.43)

Then there exists a constant C > 0 depending only on φ,H, p, and β such that

∥u′∥∞ ≤ C(φ,H, p, β)∥f∥L1(−1,1) . (3.44)

Proof. We proceed as in the proof of Theorem 3.16 and arrive at (3.40). Then for u′(1) ≥ 0, as in
the proof of Theorem 3.16 we obtain (3.41), that is,

|u′(x)| ≤ 1

minx∈[−1,1] D(x)

(
2 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) . (3.45)

For u′(1) < 0, as in the proof of Theorem 3.16 we obtain (3.42), that is,

|u′(x)| ≤ 2

minx∈[−1,1] D(x)

(
1 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) . (3.46)

Under assumption (3.43), we have

D(x) ≥ K ′ (φ,H, p, ∥f∥L1(−1,1)

)
> 0 (3.47)

by estimate (3.33) from Lemma 3.13. Using (3.45), (3.46) and (3.47), we find that

|u′(x)| ≤ 2

K ′
(
φ,H, p, ∥f∥L1(−1,1)

)
(
1 +

D(1)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) . (3.48)

Now, using the boundary condition u(1) = 0, triangle inequality, xs < x provided x > 1 for
0 < s < 1 and (a+ b)s ≤ 2s−1(as + bs) provided a,b ≥ 0 for s ≥ 1, and Lemma 3.17, we obtain

0 < D(1) = (u(1) +H)(p− 1)

∫ 1

0

|sinφ+ θu′(1) cosφ|p−2
dθ cosφ

≤ (p− 1)Hmax{1, 2p−1}
∫ 1

0

(
| sinφ|p−2 + |θu′(1) cosφ|p−2

)
dθ cosφ

= (p− 1)Hmax{1, 2p−1}
(
(sinφ)p−2 +

|u′(1) cosφ|p−2

p− 1

)
cosφ

≤ (p− 1)Hmax{1, 2p−1} cosφ(sinφ)p−2 + 2max{1, 2p−1} (cosφ)
p−2

(sinφ)p−2
∥f∥L1(−1,1) .

(3.49)

Setting

C ′(φ,H, p, β)
def
= (p− 1)Hmax{1, 2p−1} cosφ(sinφ)p−2 + 2max{1, 2p−1} (cosφ)

p−2

(sinφ)p−2
β

and taking into account that

0 < K ′ (φ,H, p, β) ≤ K ′ (φ,H, p, ∥f∥L1(−1,1)

)
,

we deduce from (3.48) and (3.49) that

|u′(x)| ≤ 2

K ′ (φ,H, p, β)

(
1 +

C ′(φ,H, p, β)

H cosφ(sinφ)p−2

)
∥f∥L1(−1,1) . (3.50)

This establishes a priori bound (3.44). □



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 23

3.6. Weak and strong maximum principles via linearization. In this section, we derive
Weak and Strong Maximum Principles for (2.8). Let us recall that, for u ∈ W 1,p

0 (−1, 1), we have

u+ def
= max{u, 0} , u− def

= max{−u, 0}
also u−, u+ ∈ W 1,p

0 (−1, 1) (see [70, Coroll. 2.1.8, p. 47]).
It is inherent to assume that f ≥ 0 a.e. when studying Weak or Strong Maximum Principle.

Let us point out that the assumption f ≥ 0 a.e. implies that Hypothesis (A4) is satisfied.
Note that even Weak Maximum Principle for the problem (2.8) is not as straightforward to prove

as in the case of the classical p-Laplacian problem, where one can use u− as a test function in the
weak formulation to prove the result. Here, the problem is caused by the term u(x) +H, which
multiplies the part depending on the derivative of the solution. This causes serious difficulty
in finding a suitable choice of test function, which would lead to conclusion. Fortunately, the
linearization method from Section 3.4 can be used under certain rather general assumptions. This
process yields the following statement.

Proposition 3.19 (Weak Maximum Principle). Let p > 2, f ∈ L1(−1, 1), and u ∈ W 1,p
0 (−1, 1)

be a weak solution to (2.8). If f ≥ 0 a.e. in (−1, 1), then u ≥ 0 in (−1, 1).

Proof. Let u ∈ W 1,p
0 (−1, 1) be any weak solution to (2.8). Then once this solution is fixed, we

define function D(·) by (3.26). Now, by Lemma 3.12, this same solution u satisfies (3.29), that is,
∫ 1

−1

D(x)u′(x)v′(x) dx+ (sinφ)p−1

∫ 1

−1

u(x)v′(x) dx =

∫ 1

−1

f(x)v(x) dx

for all v ∈ W 1,p
0 (−1, 1). Taking v = u− we have

−
∫ 1

−1

D(x)
(
[u−(x)]′

)2
dx− (sinφ)p−1

∫ 1

−1

1

2

[(
u−(x)

)2]′
dx =

∫ 1

−1

f [u(x)]− dx ≥ 0 . (3.51)

Since
D(x) ≥ K

(
φ,H, p, ∥u−∥∞

)
> 0

for all x ∈ [−1, 1] by Lemma 3.12 and
∫ 1

−1

1

2

[(
u−(x)

)2]′
dx =

1

2

[(
u−(1)

)2 −
(
u−(−1)

)2]
= 0

by the boundary conditions u(−1) = 0 = u(1), we infer from (3.51) that

K
(
φ,H, p, ∥u−∥∞

) ∫ 1

−1

([
u−(x)

]′)2

dx ≤
∫ 1

−1

D(x)
([

u−(x)
]′)2

dx ≤ 0 .

Hence [u−]′ = 0 in (−1, 1) and consequently u− = 0 in (−1, 1) since u−(−1) = 0 = u−(1) (see,
e.g., [70, Coroll. 2.1.9, p. 47]). Thus u = u+ ≥ 0 in (−1, 1). □

As a useful consequence of Weak Maximum Principle, we obtain the following result.

Corollary 3.20. Let p > 2, f ∈ L1(−1, 1), and f ≥ 0. Let u ∈ W 1,p
0 (−1, 1) be a weak solution to

(2.8). Then

D(x) ≥ 1

2
H(sinφ)p−2 cosφ > 0 (3.52)

for all x ∈ [−1, 1].

Proof. By Proposition 3.19, u ≥ 0. Hence, we can use M = H in (3.30). This establishes
(3.52). □

SinceD(·) ∈ C[−1, 1] by Lemma 3.12, we obtain from (3.52) that the function 1/D(·) ∈ C[−1, 1]
and hence it is integrable.

The crucial part of our proof of the validity of Strong Maximum Principle is to show that
Green’s function associated with the operator

Lu(x) def
= − (D(x)u′(x))

′ − (sinφ)p−1u′(x)



24 P. GIRG, L. KOTRLA EJDE-2025/57

is positive. At first, we prove that

G(x, x0)
def
=





E+(x0)−1
E−(x0)−E+(x0)

1
(sinφ)p−1 (E

−(x)− 1) for x ∈ [−1, x0] ,

E−(x0)−1
E−(x0)−E+(x0)

1
(sinφ)p−1 (E

+(x)− 1) for x ∈ (x0, 1] ,
(3.53)

x0 ∈ [−1, 1], is indeed the Green’s function associated with L. Here,

E−(s)
def
= exp

[
−

∫ s

−1

(sinφ)p−1

D(ξ)
dξ

]
,

E+(s)
def
= exp

[ ∫ 1

s

(sinφ)p−1

D(ξ)
dξ

]
.

Note that both E−(s) and E+(s) make sense since D(ξ) ≥ const. > 0 for ξ ∈ [−1, 1] by Corol-
lary 3.20 and it is continuous by Lemma 3.12. Moreover,

E+(s)− E−(s) = exp
[ ∫ 1

s

(sinφ)p−1

D(ξ)
dξ

](
1− exp

[
−
∫ 1

−1

(sinφ)p−1

D(ξ)
dξ

])

≥
(
1− exp

[
−

∫ 1

−1

(sinφ)p−1

D(ξ)
dξ

])
= const. > 0 .

(3.54)

The following result is very usefull in establishing that G is Green’s function associated with L.
Lemma 3.21. The function G : [−1, 1] × [−1, 1] → R given by (3.53) is continuous and satisfies
that

(A5) there exists κ > 0 such that |G(s, y)−G(t, y)| ≤ κ|s− t| for all s, t, y ∈ [−1, 1].

Moreover, the partial derivative Gx(x, y) exists for all (x, y) ∈ (−1, 1)× (−1, 1) such that x ̸= y.

Proof. It is easy to see from (3.53) that G ∈ C([−1, 1] × [−1, 1]) and that the partial derivative
Gx = ∂G/∂x exists for all (x, y) ∈ (−1, 1) × (−1, 1) such that x ̸= y. Then it is straightforward
to verify that G ∈ C1(Ω1) and G ∈ C1(Ω2), where

Ω1
def
= {(x, y) ∈ R : − 1 < x < 1 and 0 < y < x} , (3.55)

Ω2
def
= {(x, y) ∈ R : − 1 < x < 1 and x < y < 1} . (3.56)

Thus, for all y ∈ [−1, 1] fixed, the function G(·, y) is absolutely continuous (since continuous gluing
of two absolutely continuous functions is absolutely continuous). Now, let us set

κ = max
i=1,2

(
max

(x,y)∈Ωi

|Gx(x, y)|
)
. (3.57)

Then |Gx(x, y)| ≤ κ for all x, y ∈ [−1, 1] such that x ̸= y. Now let us consider y ∈ [−1, 1] arbitrary
but fixed. Absolute continuity of G(·, y) together with the fact that |Gx(·, y)| < κ a.e. in [−1, 1]
imply that G(·, y) is Lischitz continuous with constant κ defined by (3.57). As κ is independent
of y ∈ [−1, 1], we established the statement of the lemma. □

Theorem 3.22 (Green’s function). Let f satisfy (A4). The function

u(x) =

∫ 1

−1

G(x, y)f(y) dy

is weak solution to the linear problem (3.28).

Proof. Since

G(−1, y) =
E+(y)− 1

E−(y)− E+(y)

1

(sinφ)p−1

(
e0 − 1

)
= 0,

G(1, y) =
E−(y)− 1

E−(y)− E+(y)

1

(sinφ)p−1

(
e0 − 1

)
= 0



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 25

for all y ∈ (−1, 1), we have

u(−1) =

∫ 1

−1

G(−1, y)f(y) dy = 0,

u(1) =

∫ 1

−1

G(1, y)f(y) dy = 0 .

Thus u satisfies the boundary conditions.
Let x ∈ (−1, 1) be arbitrary. Then

du

dx
(x) =

d

dx

∫ 1

−1

G(x, y)f(y) dy

= lim
△x→0

∫ 1

−1
G(x+△x, y)f(y) dy −

∫ 1

−1
G(x, y)f(y) dy

△x

= lim
△x→0

∫ 1

−1

G(x+△x, y)−G(x, y)f(y)

△x
dy

(3.58)

Let us note that, for any fixed x ∈ (−1, 1), the partial derivative

Gx(x, y) = lim
△x→0

G(x+△x, y)−G(x, y)

△x

is well defined for any y ∈ (−1, 1) \ {x} by Lemma 3.21. Now taking into account (A5), we can
use function κ|f | ∈ L1(−1, 1) as an integrable majorant for the integrand in (3.58). Then, by the
Lebesgue dominated convergence theorem, we obtain

lim
△x→0

∫ 1

−1

(G(x+△x, y)−G(x, y)f(y)

△x
dy

=

∫ 1

−1

lim
△x→0

(G(x+△x, y)−G(x, y)f(y)

△x
dy =

∫ 1

−1

Gx(x, y)f(y) dy .

(3.59)

Thus, we obtain

du

dx
(x) =

∫ 1

−1

Gx(x, y)f(y) dy (3.60)

for all x ∈ (−1, 1). Now, using formulas for G from (3.53) and taking them into derivative, we
obtain

du

dx
(x) =

E+(x)

D(x)

∫ x

−1

E−(y)− 1

E+(y)− E−(y)
f(y) dy +

E−(x)

D(x)

∫ 1

x

E+(y)− 1

E+(y)− E−(y)
f(y) dy , (3.61)

where we used

dE±

ds
(s) = − (sinφ)p−1

D(s)
E±(s) .

Let us note that the function on the right-hand side of (3.61) can be continuously extended to
[−1, 1] as its limits exist and are finite on both ends of the interval. Thus u ∈ C1[−1, 1] and hence

its weak and strong derivatives coincide and u ∈ W 1,p
0 (−1, 1).

From (3.61), we obtain

D(x)
du

dx
(x) = E+(x)

∫ x

−1

E−(y)− 1

E+(y)− E−(y)
f(y) dy + E−(x)

∫ 1

x

E+(y)− 1

E+(y)− E−(y)
f(y) dy (3.62)

for all x ∈ (−1, 1). Taking into account (3.54), and definitions of E±, and the fact that f ∈
L1(−1, 1), we deduce that the integrands on the right-hand side of (3.62) are from L1(−1, 1).
Thus the continuous extension of D(·) du/dx to [−1, 1] belongs to AC[−1, 1]. Hence the derivative



26 P. GIRG, L. KOTRLA EJDE-2025/57

of D(·) du/dx exists a.e. in (−1, 1) and can be obtained by a straightforward calculation (as it
contains products and sums of functions from AC[−1, 1]). Indeed, we obtain

− d

dx

(
D(x)

du

dx
(x)

)
=

E−(x)− 1

E−(x)− E+(x)
E+(x)f(x)− E+(x)− 1

E−(x)− E+(x)
E−(x)f(x)

+
(sinφ)p−1E+(x)

D(x)

∫ x

−1

E−(y)− 1

E+(y)− E−(y)
f(y) dy

+
(sinφ)p−1E−(x)

D(x)

∫ 1

x

E+(y)− 1

E+(y)− E−(y)
f(y) dy

(3.63)

a.e. in (−1, 1). Using (3.61), we obtain

(sinφ)p−1 du

dx
(x) =

(sinφ)p−1E+(x)

D(x)

∫ x

−1

E−(y)− 1

E+(y)− E−(y)
f(y) dy

+
(sinφ)p−1E−(x)

D(x)

∫ 1

x

E+(y)− 1

E+(y)− E−(y)
f(y) dy .

(3.64)

Subtracting (3.64) from (3.63), we see that

− d

dx

(
D(x)

du

dx
(x)

)
− (sinφ)p−1 du

dx
(x)

=
E−(x)− 1

E−(x)− E+(x)
E+(x)f(x)− E+(x)− 1

E−(x)− E+(x)
E−(x)f(x)

=
E−(x)E+(x)− E+(x)− E−(x)E+(x) + E−(x)

E−(x)− E+(x)
f(x) = f(x)

for almost every x ∈ (−1, 1). Now, let us recall that the continuous extension of D(·) du/dx to

[−1, 1] belongs to AC[−1, 1]. Thus, for any v ∈ W 1,p
0 (−1, 1) (working with its representative in

AC[−1, 1]), the following integration by parts makes sense

∫ 1

−1

D(x)u′(x)v′(x)dx+ (sinφ)p−1

∫ 1

−1

u(x)v′(x)dx

=

∫ 1

−1

(
(D(x)u′(x))′v(x) + (sinφ)p−1u(x)′

)
v(x)dx

=

∫ 1

−1

f(x)v(x)dx ,

(3.65)

where we use a shorter notation for derivatives. This establishes that the function u(x) =∫ 1

−1
G(x, y)f(y) dy is a weak solution to the linear problem (3.28). □

Lemma 3.23 (Positivity of Green’s funtion). The Green’s function of L satisfies G(x, y) > 0 for
all x, y ∈ (−1, 1).

Proof. Taking into account that D(ξ) ≥ const. > 0, we have

0 < E−(s) = exp
[
−

∫ s

−1

(sinφ)p−1

D(ξ)
dξ

]
< 1,

1 < E+(s) = exp
[ ∫ 1

s

(sinφ)p−1

D(ξ)
dξ

]

for all s ∈ (−1, 1). Then the positivity of Green’s function given by (3.53) follows from the
inequalities above combined with inequality (3.54). □

Theorem 3.24 (Strong Maximum Principle). Let u ∈ W 1,p
0 (−1, 1) be a solution to (2.8) with

f ∈ L1(−1, 1). If f ≥ 0 and f ̸≡ 0, then u > 0 on (−1, 1).



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 27

Proof. We showed in Section 3.4, that any solution u to nonlinear problem (2.8) is also solution to
linear problem (3.28) with spatially dependent diffusion coefficient D(·) constructed from u. By
our assumption there exists measurable set A ⊂ (−1, 1) of positive Lebesgue measure such that
f > 0 a.e. on A and, moreover, f ≥ 0 a.e. in (−1, 1). Hence by positivity of Green’s function
obtained in Lemma 3.23 and by Theorem 3.22, we have

u(x) =

∫ 1

−1

G(x, y)f(y) dy ≥
∫

A

G(x, y)f(y) dy > 0 ,

for any x ∈ (−1, 1). □

4. Concluding remarks

Table 1. Main results.

Result Assumptions on f p Result type

Thm. 3.6 (A4) p > 1 C1-regularity.

Thm. 3.8 (A4) p > 1 A priori bound on ∥u∥∞.

Thm. 3.9 ∥f∥L1(−1,1) < H(sinφ)p−1 p > 1 Existence of weak solution.

Lem. 3.12 (A4) p > 2 Linearization.

Thm. 3.16 (A4) p > 2 Bound on ∥u′∥∞.

Thm. 3.18
(A4);

∥f∥L1(−1,1) ≤ β < H(sinφ)p−1 p > 2 A priori bound on ∥u′∥∞.

Prop. 3.19 f ∈ L1(−1, 1); f ≥ 0 p > 2 Weak Max. Principle.

Thm. 3.24 f ∈ L1(−1, 1); f ≥ 0; f ̸≡ 0 p > 2 Strong Max. Principle.

We proposed a new mathematical model of groundwater flow over inclined impermeable bed. It
turned out that this lead us to a strongly nonlinear problem, which is quite difficult to investigate.
For summary of our main results, see Table 1 together with their assumptions on f and p.

The most important of our qualitative results cover the case p > 2 only, which corresponds to
flow in media of low permeability such as certain sandstones, fine sands, clays or certain types of
soils (see, e.g., [39, 59, 71]). Let us note that the case 3/2 < p < 2 is very important in applications
too, as it corresponds to flows in coarse grained porous media such as gravels, see, e.g., [10, 59, 71].
Thus generalizations of results valid for p > 2 stated in Table 1 to include all p > 3/2 would be of
great importance. For further discussion, see Remark 3.15.

Let us also note that f ∈ L1(−1, 1) and f ≥ 0 imply (A4), so we do not explicitly assume
(A4) in results concerning maximum principles, where f ≥ 0 is natural assumption. We did not
tackle Weak and Strong Comparison Principles, which for nonlinear problems are more difficult to
prove than maximum principles. We suggest an investigation of comparison principles as a very
promising direction of research from both theoretical and practical point of view.

Another interesting direction of research would be to consider situations when (A4) is not
satisfied. Then the solution u can reach the value −H in some subdomains of (−1, 1). This
situation would correspond to the case that there is no groundwater over such subdomains. This
is, however, realistic scenario worth of further research. Last but not least, it is also of great
importance to study the case when H = 0, which corresponds to the case when the ditches are
dry.

Finally, let us remark that the use of modern geophysical non-invasive techniques helps to un-
derstand internal structures of slopes, see, e.g., [25, 44, 47, 64]. Using these techniques, locations
with porous media layered over impermeable beds (bedrock) have been observed in natural land-
scapes. These locations are encountered, e.g., in hilly or mountainous areas of central Europe,
which shed water into the surrounding highly populated areas and are rich in precipitation. Hence,
it is of great practical importance to understand water flow in such locations. But in these cases,



28 P. GIRG, L. KOTRLA EJDE-2025/57

boundary conditions other than Dirichlet need to be considered. Also the imaging techniques from
[25, 44, 47] reveal that the bedrock does not have to be flat hyperplane and realistic model needs
to take into account possibly curved surface of the bedrock. This suggests another interesting
direction of future research.

5. Appendix

A convenient approach how to show that the operator

A : W 1,p
0 (−1, 1) → W−1,p′

(−1, 1)

defined by the left-hand side of (3.22) has desired properties, is to recognize that (3.22) is a
specific instance of [46, Eq. (2.51), p. 44]. Then it suffices to write expressions for functions
a, c : (−1, 1) × R × R from [46, Eq. (2.51), p. 44] to match left-hand side of (3.22) and verify
structural assumptions [46, (2.54), (2.55), (2.65), and (2.92)].

Indeed, it is easy to see that the choice

a(x, r, s) = (H + Tk(r)) (cosφ)
p−1|s+ tanφ|p−2(s+ tanφ) , (5.1)

c(x, r, s) = 0 , (5.2)

in [46, Eq. (2.51), p. 44] matches the expression on the left-hand side of (3.22). Let us recall that
Tk(t) = max{−k,min{t, k}}. Since ΓN from [46, (2.54)] is ΓN = ∅ in our case, the function b
from [46, (2.54)] is set to be zero. Both functions b = 0 and c = 0 trivially satisfy all required
assumptions posed thereon in [46]. Thus we do not mention these conditions on b and c explicitly
here.

Boundedness. Taking into account that our function a is continuous in variables r and s (and
does not depend on x), it is a Carathéodory function. From the growth conditions [46, (2.55a–c)],
we need to verify only [46, (2.55a)], since the other are satisfied trivially in our situation. Taking
account that

|a(x, r, s)| =
∣∣ (H + Tk(r)) (cosφ)

p−1|s+ tanφ|p−2(s+ tanφ)
∣∣

≤ |H +max{−k,min{r, k}}| |s+ tanφ|p−1 ≤ 2p−2(H + k)
(
|s|p−1 + | tanφ|p−1

)

is satisfied for all x ∈ (−1, 1), r, s ∈ R, the condition [46, (2.55a)] is verified. According to

[46, Lemma 2.31], conditions [46, (2.54) and (2.55)] ensure that the operator A : W 1,p
0 (−1, 1) →

W−1,p′
(−1, 1) is bounded.

Coercivity. Observe that

lim
s→±∞

|s+ tanφ|p−2(s+ tanφ) s

|sp| = 1

for any p > 1 and φ ∈ (0, π/2). Hence

lim
s→±∞

|s+ tanφ|p−2(s+ tanφ) s− 1

2
|s|p = +∞

and, by continuity argument,

µ
def
= −min

{
0,min

s∈R
|s+ tanφ|p−2(s+ tanφ) s− 1

2
|s|p

}
≥ 0

exists and is finite. Then

|s+ tanφ|p−2(s+ tanφ) s =
1

2
|s|p + |s+ tanφ|p−2(s+ tanφ) s− 1

2
|s|p

≥ 1

2
|s|p +min

s∈R

(
|s+ tanφ|p−2(s+ tanφ) s− 1

2
|s|p

)

≥ 1

2
|s|p − µ

(5.3)



EJDE-2025/57 FLOW IN POROUS MEDIUM LAYERED OVER INCLINED IMPERMEABLE BEDS 29

For 0 < k < H, H + Tk(r) ≥ H − k > 0. Thus

a(x, r, s)s = (H + Tk(r)) (cosφ)
p−1|s+ tanφ|p−2(s+ tanφ)s

≥ (H − k) (cosφ)p−1|s+ tanφ|p−2(s+ tanφ)s

≥ 1

2
(H − k) (cosφ)p−1|s|p − (H − k)) (cosφ)p−1µ ,

(5.4)

which means that the structural condition [46, (2.92a)] is verified. According to [46, Lem. 2.35],

the operator A : W 1,p
0 (−1, 1) → W−1,p′

(−1, 1) is coercive.

Pseudomonotonicity. To prove this we will use the following well-known results.

Proposition 5.1 (see [55, pp. 210–211]). Let p > 1 and N ∈ N. Then there exists a constant
cp > 0 (depending on p) such that, for all x, y ∈ RN , the following inequality holds

⟨|x|p−2x− |y|p−2y, x− y⟩RN ≥
{
cp|x− y|p if p ≥ 2 ,

cp
|x−y|2

(|x|+|y|)2−p if 1 < p < 2 .

Now we are ready to verify condition [46, Eq. (2.65)] (the so called monotonicity in the main
part). For this we introduce new variables σ = s− tanφ, σ̃ = s̃− tanφ and observe s− s̃ = σ− σ̃.
With this, we have

(a(x, r, s)− a(x, r, s̃))(s− s̃)

(a(x, r, σ + tanφ)− a(x, r, σ̃ + tanφ))(σ − σ̃)

(H + Tk(r)) (cosφ)
p−1

(
|σ|p−2(σ)− |σ̃|p−2(σ̃)

)
(σ − σ̃)

≥
{

cp (H + Tk(r)) (cosφ)
p−1|σ − σ̃|p if p ≥ 2

cp (H + Tk(r)) (cosφ)
p−1 |σ−σ̃|2

(|σ|+|σ̃|)2−p if 1 < p < 2

}
≥ 0 ,

for all x ∈ (−1, 1), r, s, s̃ ∈ R. Now, by [46, Lem. 2.32], conditions [46, (2.54), (2.55), and (2.65)]

ensure that the operator A : W 1,p
0 (−1, 1) → W−1,p′

(−1, 1) is pseudomonotone.

In memory of Prof. Neuberger’s Legacy of inspiration and guidance

As a young researcher attending the “Variational Methods: Open Problems, Recent Progress,
and Numerical Algorithms” conference held in Flagstaff AZ, USA, in 2002, Petr Girg had the
privilege of meeting Professor John W. Neuberger and engaging in several stimulating discussions
that brought to his attention a deeper connection between theoretical, applied, and numerical
mathematics. These discussions had a significant impact on Petr’s later career, and he would like
to take this opportunity to express his sincere gratitude for Professor Neuberger’s insightful advice
and encouragement. Professor Neuberger’s absence will be deeply felt by younger researchers
across the field of mathematics.

Acknowledgements. P. Girg and L. Kotrla were supported by the Grant Agency of the Czech
Republic, Grant No. 22-18261S.

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32 P. GIRG, L. KOTRLA EJDE-2025/57

Petr Girg

Department of Mathematics and NTIS, Faculty of Applied Scences, University of West Bohemia, Uni-

verzitńı 8, CZ-301 00 Plzeň, Czech Republic
Email address: pgirg@kma.zcu.cz

Lukáš Kotrla

Department of Mathematics and NTIS, Faculty of Applied Scences, University of West Bohemia, Uni-
verzitńı 8, CZ-301 00 Plzeň, Czech Republic

Email address: kotrla@ntis.zcu.cz


	1. Introduction
	2. Mathematical model of water flow in porous medium layered over an inclined impermeable bed
	2.1. Initial physical considerations
	2.2. Constitutive law
	2.3. Free surface problem and Dupuit-Forchheimer assumption
	2.4. Dupuit-Forchheimer assumption on the sloping porous medium
	2.5. Physical assumptions of our model
	2.6. Initial-boundary value problem for a mathematical model of groundwater flow between infinite parallel ditches

	3. Properties of weak solution
	3.1. Regularity results
	3.2. A priori bound on a weak solution
	3.3. Existence of a weak solution
	3.4. Linearization at the trivial solution.
	3.5. Additional regularity results.
	3.6. Weak and strong maximum principles via linearization

	4. Concluding remarks
	5. Appendix
	In memory of Prof. Neuberger's Legacy of inspiration and guidance
	Acknowledgements

	References