EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 4050-4058
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

On L∞ and L2 Bounds for Weak Solutions of Time
Flows for Certain Functionals of Linear Growth

Thomas Wunderli

Department of Mathematics and Statistics, The American University of Sharjah, Sharjah,
United Arab Emirates

Abstract. We use recent approximation results in BV space to derive L∞ bounds for u, L∞

bounds in time for the BV seminorm
∫
|Du| of u, and L2 bounds for ut for the weak solution

u ∈ C([0,∞);L2 (Ω) ∩BV (Ω)), Ω ⊂ RN open and bounded, of the time flow

∂u

∂t
= div∇pφ(x,Du)− λ(u− u0), λ > 0, u(0, x) = u0.

We assume Neumann boundary condition and φ(x, p) is in a class of linear growth functions in
p. Importantly, φ(·, p) ∈ L1 (Ω) in contrast to the classical results stated in [1] where φ has a
continuity assumption in the x variable. We also use the convergence of the solution above to
derive an L∞ bound for the solution u∗ to the corresponding stationary problem, since u(t) → u∗

in L1 (Ω) .

2020 Mathematics Subject Classifications: 49Jxx, 35D30

Key Words and Phrases: Bounded variation, weak solution, variational problems, linear growth

1. Introduction

The theory of existence and qualitative properties on bounded, open Ω ⊂ RN of time
flow problems of the form

∂u

∂t
= div∇pg(x,Du) (1)

where u ∈ L2 ((0, T ) : BV (Ω) ∩ L2 (Ω)) with initial data u(0, ·) = u0 ∈ L2(Ω), boundary
data u = h on ∂Ω, and g(x, p) convex in p with linear growth in p has been covered
and summarized extensively in [1]. Since for each t, u(t, ·) ∈ BV (Ω) and that W 1(Ω) ⊊
BV (Ω) the divergence term on the right of the equation is not well defined. The solution
has to be defined in the context of nonlinear semigroup theory as the authors do in the
collection of results in [1]. In fact it is proved there that there is a solution to (1) in the
sense of Definition 6.5 in [1] with initial and boundary conditions u(0, x) = u0(x) with

DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5328

Email address: twunderli@aus.edu (T. Wunderli)

https://www.ejpam.com 4050 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4051

u0 ∈ L2 (Ω), u(t, x) = h(x), and h ∈ L1 (∂Ω) . The solution u ∈ C([0, T ];L2 (Ω)∩BV (Ω)),
satisfies u(0) = u0, u

′(t) ∈ L2(Ω), and u′(t) = div∇pg(x,Du) in D′ (Ω) , that is, in the
distributional sense. It is also assumed that g is continuous on Ω × RN . Many of these
results rely on the approximation of

∫
Ω g(x,Du) for u ∈ BV (Ω) by

∫
Ω g(x,∇uk) dx for uk

smooth. However the approximation theorems assume continuity or lower semicontinuity
of g in (x, p). We also note the more recent work of [17] for time flows in BV space using
the Allen-Cahn equation.

In this work we use new approximation results (Proposition 1) to derive an L∞ bound
for u and an L2 bound for ut for the weak solution, as defined in [7] or [22], to the Neumann
problem 

∂u
∂t = div∇pφ(x,Du)− λ(u− u0) in (0,∞)× Ω, λ > 0
∂u
∂n = 0 on (0,∞)× ∂Ω
u(0, x) = u0(x) for x ∈ Ω, u0 ∈ L∞ (Ω) .

(2)

In fact, we show the weak solution to (2) satisfies u ∈ L∞ ([0,∞);BV (Ω) ∩ L∞ (Ω)),
ut ∈ L2 ((0,∞)× Ω). Importantly, while φ(x, p) is also of linear growth, convex and C2 in
p, there is no continuity assumption in the x variable with only φ(·, p) ∈ L1 (Ω) . Using the
L∞ bound, as noted in Theorem 2 below, we easily prove an L∞ bound for the solution
to the corresponding time independent minimization problem of Theorem 1.

For the integrand φ we first assume:
(1) φ : Ω× RN → R, where φ(x, p) is convex in p, that is

φ(x, λ1p1 + λ2p2) ≤ λ1φ (x, p1) + λ2φ (x, p2)

for each z ∈ R, p1, p2 ∈ RN , 0 ≤ λ1, λ2 ≤ 1, λ1 + λ2 = 1,
(2) φ(x, p) = φ(x, |p|) is radially symmetric in p, and is of the form

φ(x, p) =

{
g(x, p) if |p| ≤ β
ψ(x)|p|+ k(x) if |p| > β

(3)

for k ∈ L1 (Ω) and ψ ∈ C
(
Ω
)
.

(3) φ is a Carathéodory function, with φ(·, p) ∈ L1 (Ω) for each p.
From (2), φ is of linear growth in the p variable as in [1], that is

lim
|p|→∞

φ(x, p)

|p|
= ψ(x).

We note that time flows and functionals defined on BV include applications starting
with the early examples of total variation flow in [16] and elastic plastic deformation in
[11] and [12]. In fact, solving the time flow and letting t → ∞ for the solution u(t) gives
the solution u to the stationary problem in these cases. For example, in [6], a functional
with an integrand of the form φ as in (3) has applications to anisotropic noise removal in
image processing with the assumption φ(·, p) ∈ L∞ (Ω). The model used there is

min
u∈BV (Ω)

∫
Ω
φ(x,Du) + λ/2 ∥u− u0∥2L2(Ω)



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4052

λ > 0, where the solution u is taken to be the restored image and u0 is the noisy or
corrupted image. The integrand φ in [6] is

φ(x, p) =

{
1

r(x) |p|
r(x) if |p| ≤ 1

|p| − r(x)−1
r(x) if |p| > 1

with 1 < α ≤ r(x) ≤ 2, r ∈ L∞ (Ω) , which corresponds to k(x) = − r(x)−1
r(x) and ψ(x) ≡ 1

for (3). In addition, the authors provide numerical examples and prove existence results
for the corresponding time flow, including the convergence of the time flow solution u(t)
to u in L1 (Ω) as t→ ∞, where u is the solution to the above minimization problem. We
note that many of the results proved there are based on the specific form of φ used in [6].
In our case, we include the more general condition that φ(·, p) ∈ L1 (Ω) and that φ takes
a more general form than in [6].

2. Preliminary Results

We first recall Lemma 1 in [20]

Lemma 1. Assume φ satisfies the conditions (1)-(3) above:

φ(x, p) =

{
g(x, p) if |p| ≤ β
ψ(x)|p|+ k(x) if |p| > β,

with ψ ∈ C
(
Ω
)
, ψ ≥ 0, k(x) ∈ L1 (Ω) for each u ∈ L1 (Ω). Also assume for some G

φ(x, p) = G(r1(x), ..., rK(x), p) for all p

where

G(z1, ..., zK , p) =

{
g1(z1, ..., zK , p) if |p| ≤ β

zK |p|+ g2(z1, ..., zK) if |p| > β

and where for each |p| ≤ β, g1 is C1 in the variable z = (z1..., zK) ∈ U ⊂ RK , U open,
ri ∈ L1 (Ω) each i, (r1(x), ..., rK(x)) ∈ U a.e. x, and |(∇zg1)(z, p)| ≤ C, C independent of
(z, p). Note that rK(x) = ψ(x) and hence zk ≥ 0.

Then for all u ∈ BV (Ω) we have

G(u) =

∫
Ω
φ(x,∇u) dx+

∫
Ω
ψ(x)|Dsu|

= sup
{ϕ∈C1

0 (Ω,RN ):|ϕ(x)|≤ψ(x) for all x∈Ω}

{
−
∫
Ω
udivϕ+ φ∗(x, ϕ(x)) dx

}
,

and hence G is lower semicontinuous in L1 (Ω) .

In order to prove the bounds for the weak solution u, we need the following proposition
to extend the approximation Lemma from [21] to include time dependence, which covers
the case where we only have φ(·, p) ∈ L1 (Ω) .



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4053

Proposition 1. If φ satisfies conditions in Lemma 1 and φ(x, p) ≥ 0 for a.e. x,
each p, then for each u ∈ L2([0, T ];BV (Ω) ∩ L2 (Ω)), there exists a sequence uk ∈
L2([0, T ];W 1,1 (Ω) ∩ C∞ (Ω) ∩ L2 (Ω)) with∫ T

0

∫
Ω
φ(x,Duk) dxdt →

∫ T

0

∫
Ω
φ(x,Du) dt and

uk → u in L2 ([0, T ]× Ω) .

If ∂Ω is Lipschitz, we can choose uk ∈ L2([0, T ];C∞ (
Ω
)
).

Proof. We follow the proof in [21] (also see [9], [10] for the pure total variation case)
with the same partition of unity Ωi for Ω resulting in the partition {[0, T ]× Ωi} , with
the standard smoothing

(ηε ∗ u)(t, x) =
∫
Bε(x)

ηε(x− y)u(t, y) dy

in the x variable only. Noting that each [0, T ]×support(ϕi) is compact, we choose
1. each 0 < εi < ε, i ≥ 1
2.

∫ T
0

∫
Ω |ηεi ∗ (uϕi)− uϕi|2 dx ≤ ε2−i

3.
∫ T
0

∫
Ω |ηεi ∗ (u∇ϕi)− u∇ϕi| dx ≤ ε2−i

4. support ηεi ∗ (uϕi) ⊂ [0, T ]× Ωi+2 − [0, T ]× Ωi−2.
Then for uε defined by uε =

∑∞
i=1 ηεi ∗ (uϕi) we have uε → u in L2([0, T ]×Ω). Passing

to a subsequence of ε we have uε → u in L2 (Ω) for a.e. t. Thus for a.e. t,∫
Ω
φ(x,Du) ≤ lim inf

ε→0

∫
Ω
φ(x,Duε) dx.

Since φ(x, p) ≥ 0, by Fatou’s Lemma we have∫ T

0

∫
Ω
φ(x,Du) ≤ lim inf

ε→0

∫ T

0

∫
Ω
φ(x,Duε) dx. (4)

For the above subsequence in ε, proceed as in the proof there to get for a.e. t, after
taking the supremum over relevant ϕ ∈ C1

0 (Ω,RN ) with |ϕ(x)| ≤ ψ(x) for each x∫
Ω
φ(x,Duε) ≤

∫
Ω
φ(x,Du) +

∫
Ω
ω(ε1)|∇u| dx

+ω(ε1)

∫
Ω
d|Dsu|+ 2β|ψ|∞ε

+(sup
ϕ
II + sup

ϕ
|III|+ sup

ϕ
|IV |+ ω(ε1)|ψ|∞ |Ω|),

where ω is a modulus of continuity for ψ with ω(t) → 0 as t→ 0+ and II, III, IV are the
same terms as in [21]. From the proof of the approximation Lemma in [21] and [10] we

have
∫ T
0 supϕ II dt→ 0 as ε→ 0 since u ∈ L2([0, T ];BV (Ω) ∩L2 (Ω)),

∫ T
0 supϕ |III | dt ≤



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4054

|ψ|∞εT from item 3, and
∫ T
0 supϕ |IV | ≤ (βε+2β|ψ|∞ε)T. Now integrate with respect to

t to get∫ T

0

∫
Ω
φ(x,Duε) dt ≤

∫ T

0

∫
Ω
φ(x,Du) dt+

∫ T

0

∫
Ω
ω(ε1)|∇u| dxdt

+ω(ε1)

∫ T

0

∫
Ω
d|Dsu| dt+ 2β|ψ|∞εT +

∫ T

0
sup
ϕ
II dt

+[ε|ψ|∞ + βε+ 2β|ψ|∞ε+ ω(ε1)|ψ|∞ |Ω|]T.

Send ε→ 0 to obtain

lim sup
ε→0

∫ T

0

∫
Ω
φ(x,Duε) dxdt ≤

∫ T

0

∫
Ω
φ(x,Du).

This combined with (4) proves the first part.
If ∂Ω is Lipschitz, using the fact that L2([0, T ];C∞ (

Ω
)
) is dense in L2([0, T ];W 1,1 (Ω)∩

L2 (Ω)) (from a simple modification of Theorem 3, section 4.2 in [9]), a modification of
Remark 2.2.8 in [7] and by noting from Lemma 1 in [18] that∫ T

0

∫
Ω
|φ(x,∇v)− φ(x,∇u)| dxdt ≤ ∥ψ∥∞

∫ T

0

∫
Ω
|∇v −∇u| dxdt

for each u, v ∈ L2([0, T ];W 1,1 (Ω)), we can choose uk ∈ L2([0, T ];C∞ (
Ω
)
).

Remark 1. We note that the assumption ψ ∈ C
(
Ω
)
is used here so that ψ is uniformly

continuous, as the original assumption of ψ ∈ C (Ω) ∩ L∞(Ω) in Lemma from [21] was
incorrect.

3. Bounds for the Weak Solution

We recall the definition of a weak solution as used [22] or [7] for the time flow problem.

Definition 1. We define the weak solution u ∈ L2([0,∞);BV (Ω) ∩ L2 (Ω)) of the initial
value Neumann problem

∂u
∂t = div∇pφ(x,Du)− λ(u− u0) in (0,∞)× Ω, λ > 0
∂u
∂n = 0 on (0,∞)× ∂Ω
u(0, x) = u0(x) for x ∈ Ω, u0 ∈ L∞ (Ω)

(5)

to be the following: u ∈ L2([0,∞) : BV (Ω) ∩ L2 (Ω)) with ut :=
∂u
∂t ∈ L2(Ω× [0,∞)) is a

weak solution of (5) if∫ s

0

∫
Ω
ut(v − u) dxdt+

∫ s

0

∫
Ω
φ(x,Dv) dt+

∫ s

0

∫
Ω
(v − u0)

2 dxdt ≥ (6)∫ s

0

∫
Ω
φ(x,Du) dt+

∫ s

0

∫
Ω
(u− u0)

2 dxdt

for all v ∈ L2([0,∞) : BV (Ω) ∩ L2 (Ω)) for a.e. s ∈ [0,∞).



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4055

We now assume φ satisfies the coercivity condition (4) φ(x, p) ≥ c|p|, c > 0, for a.e. x,
each p.

We first note that the existence of a semigroup solution u ∈ C([0,∞);L2 (Ω)) with
ut ∈ L∞((0,∞);L2 (Ω)) is guaranteed by the standard theory of nonlinear semigroups for
maximal monotone operators since the functional

Φ(u) :=

{ ∫
Ω φ(x,Du) +

λ
2

∫
Ω(u− u0)

2 dx, λ > 0, for u ∈ BV (Ω) ∩ L2 (Ω)
∞ for u ∈ L2 (Ω) \BV (Ω)

(7)

is convex and lower semicontinuous on L2 (Ω) due to Lemma 1, and is hence a maximal
monotone operator. We also have

Theorem 1. If φ satisfies the condition of Lemma 1 and the coercivity condition (4) ,
then problem

min
u∈BV (Ω)∩L2(Ω)

Φ(u)

for Φ defined by (7) has a unique solution.

Proof. This follows from standard results due to coercivity, lower semicontinuity of
Φ(u) in L2 (Ω) from Lemma 1, compactness of BV, and strict convexity of Φ.

The semigroup solution u(t) for (7) satisfies

u(0) = u0

u(t) ∈ D(∂Φ) for each t > 0

−u′(t) ∈ ∂Φ[u(t)] for a.e. t ≥ 0,

where ∂Φ is the subdifferential of Φ (see for example [5], [8]). From the definition of ∂Φ
it follows ∫

Ω
ut(v − u(t)) dx +Φ(v) ≥ Φ(u(t)) for a.e. t ≥ 0

for each v ∈ L2(Ω). We again note that necessity of lower semicontinuity of the Φ(u) term.
For other recent cases where lower semicontinuity holds for functions defined on BV , see
for example [2], [3], [13], [14], and [15].

In Theorem 2 we additionally prove u ∈ L∞ ([0,∞);BV (Ω) ∩ L∞ (Ω)), ut ∈ L2 ((0,∞)× Ω)
for the weak solution given in Definition 1.

Theorem 2. If φ satisfies the assumptions of Lemma 1, the coercivity condition (4) ,
is C2 in p, φ(x, p) ≥ φ(x, 0) a.e. x for all p and ∂Ω Lipschitz, then there exists a weak
solution u to (5) where u ∈ L∞ ([0,∞);BV (Ω) ∩ L∞ (Ω)) , ut ∈ L2 ((0,∞)× Ω) and∫ ∞

0

∫
Ω
(ut)

2 dxdt+

∫
Ω
φ(x,Du) ≤

∫
Ω
φ(x,Du0) for a.e. t ∈ [0,∞)

∥u∥L∞([0,∞)×Ω) ≤ C (Ω) ∥u0∥∞ .



T. Wunderli / Eur. J. Pure Appl. Math, 17 (4) (2024), 4050-4058 4056

Proof. The proof essentially follows the earlier works of [6], [7], or [22], by considering
the solution uεδ ∈ L2

(
[0,∞);H1 (Ω)

)
to the approximation problem

∂u
∂t = ε∆u+ div∇pφ(x,∇u)− λ(u− uδ0) in [0, T ]× Ω
∂u
∂n = 0 on [0, T ]× ∂Ω

u(0, x) = uδ0(x) for x ∈ Ω, uδ0 ∈ BV (Ω) ∩ C∞ (
Ω
)
.

We use the fact that uεδ satisfies the form of the weak solution given in (6) with φ replaced
by φ(x, p)+ ε

2 |p|
2 and v ∈ L2([0,∞);H1 (Ω)), passing to limits ε→ 0, δ → 0 after obtaining

the appropriate L∞ and L2 bounds, and finally using the Lipschitz assumption of ∂Ω and
Proposition 1 to get (6) for v ∈ L2([0,∞) : BV (Ω) ∩ L2 (Ω)).

From the works cited in the proof of Theorem 2 above, the weak solution also satisfies∫ s

0

∫
Ω
ut(v − u) dxdt+

∫ s

0

∫
Ω
φ(x,Dv) dt ≥ (8)∫ s

0

∫
Ω
φ(x,Du) dt− λ

∫ s

0

∫
Ω
(u− u0)(v − u) dxdt.

It is then straightforward to show if u is a weak solution of (5) then for each t > 0∫
Ω
ut(v − u) dxdt+

∫
Ω
φ(x,Dv) dt ≥ (9)∫

Ω
φ(x,Du) dt− λ

∫
Ω
(u− u0)(v − u) dxdt

and hence using Young’s inequality for the last term on the right∫
Ω
ut(v − u) dxdt+

∫
Ω
φ(x,Dv) dt+

∫
Ω
(v − u0)

2 dxdt ≥ (10)∫
Ω
φ(x,Du) dt+

∫
Ω
(u− u0)

2 dxdt for each v ∈ BV (Ω) ∩ L2 (Ω) .

Thus u also a semigroup solution. Letting v = u+ λϕ for ϕ ∈ C∞
c (Ω) in (10) and letting

λ→ 0+ and λ→ 0− we have

Corollary 1. If u is a solution to (6) then we have

∂u

∂t
= div∇pφ(x,∇u)− λ(u− u0) in [0, T ]× Ω

in D′ (Ω) .

4. Conclusion

In this work we proved L∞ and L2 bounds for weak solutions in BV for time flows
(2) of the minimization problem from Theorem 1 for a class of integrands φ(·, p) ∈ L1(Ω);
whereas most of the previous results include a continuity assumption in x. For future
consideration, we may consider integrands φ that are not C2 in the variable p as well as
more general integrands g that are not specifically of the form φ as stated in this work,
but with g(·, p) ∈ L1(Ω) and g convex p.



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