EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2025-2034
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Approximation of BV space-defined functionals
containing piecewise integrands with L1 condition

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

Abstract. We prove an approximation result for a class of functionals G(u) =
∫
Ω
φ(x,Du) defined

on BV (Ω) where φ(·, Du) ∈ L1 (Ω) , Ω ⊂ RN bounded, φ(x, p) convex, radially symmetric and of
the form

φ(x, p) =

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

We show for each u ∈ BV (Ω) ∩ Lp (Ω) , 1 ≤ p < ∞, there exist uk ∈ W 1,1 (Ω) ∩ C∞ (Ω) ∩ Lp (Ω)
so that G(uk) → G(u). Approximation theorems in BV are used to prove existence results for the
strong solution to the time flow ut = div (∇pφ(x,Du)) in L1((0,∞);BV (Ω) ∩ Lp (Ω)), typically
with additional boundary condition or penalty term in u to ensure uniqueness. The functions in this
work are not covered by previous approximation theorems since for fixed p we have φ(x, p) ∈ L1 (Ω)
which do not in general hold for assumptions on φ in earlier work.

2020 Mathematics Subject Classifications: 49Nxx, 35Dxx

Key Words and Phrases: Bounded variation, conjugate function, Carathéodory function, vari-
ational problems

1. Introduction

In this work, we present some approximation results for functionals

G(u) :=
∫
Ω
φ(x,Du) (1)

defined for u ∈ BV (Ω) with bounded, open Ω ⊂ RN with the following assumptions on
φ:

(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,

DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4934

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

https://www.ejpam.com 2025 © 2023 EJPAM All rights reserved.



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2026

(2) φ(x, p) = φ(x, |p|) for all p, and for k ∈ L1 (Ω) is of the form

φ(x, p) =

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

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

lim
|p|→∞

φ(x, p)

|p|
= ψ(x).

We note that φ(x, p) is continuous in p since real valued convex functions are con-
tinuous. The main result of this paper is the extension of the approximation theorems
presented in, [2], [5], and [8] to include certain cases where φ(·, p) ∈ L1 (Ω) for a class of
integrands φ with the above assumptions (1)-(3). We note that functionals of the form (1)
defined on BV have many applications to elasticity and image processing problems (see
e.g. the early works of [9], [12], [14], [19]).

We recall the classic approximation theorem in [8] where it is proved that for each
u ∈ BV (Ω), Ω ⊂ RN bounded, there exists a sequence {uk} ⊂ W 1,1 (Ω) ∩ C∞ (Ω) so
that uk → u in L1 (Ω) and

∫
Ω |∇uk| dx →

∫
Ω |Du|. We recall u ∈ BV (Ω) if and only if

u ∈ L1 (Ω) and∫
Ω
|Du| := sup

ϕ∈{C∞
0 (Ω,RN ),|ϕ(x)|≤1 all x∈Ω}

{
−
∫
Ω
udivϕ dx

}
<∞,

and with ∥u∥BV (Ω) := ∥u∥L1(Ω) +
∫
Ω |Du|. In this case we have

∫
Ω |Du| :=

∫
Ω |∇u| dx +∫

Ω |Dsu| for the measures ∇u dx << LN and Dsu ⊥ LN , and where Dsu = 0 if and only
if u ∈W 1,1 (Ω) . As W 1,1 (Ω) is not dense in BV (Ω) we can not have

∫
Ω |∇uk −Du| → 0.

See [7] for a detailed discussion.
As a model for image restoration, the authors in [5] consider

Φh(u) : =

∫
Ω
φ(x,Du) +

λ

2

∫
Ω
(u− u0)

2 dx+∫
∂Ω

|u− h| dHN−1

for λ > 0 constant, u0 ∈ L∞ (Ω), and where

φ(x, p) =

{
1

q(x) |p|
q(x) |p| ≤ β

|p| − βq(x)−βq(x)

q(x) |p| > β

for constant β > 0, q ∈ L∞ (Ω) , 1 < α ≤ q(x) ≤ 2. Here u and h are defined on ∂Ω in the
sense of trace ([7]). The solution to

min
u∈BV (Ω)

Φh(u) (2)



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2027

is then the restored version of the corrupted image u0. In order to prove the existence
of the weak solution of the corresponding time flow for (2), the authors show for each
u ∈ BV (Ω) there is a sequence uk ∈ H1 (Ω) ∩ C∞ (Ω) where

uk → u in L2 (Ω) and

Φh(uk) → Φh(u).

Other approximation results are proved in [2] (Lemma 6.2) assuming lower semicon-
tinuity or continuity in the x variable and in [3] for integrand g(x, p) with a continuity
condition in x which in general will not be satisfied in our case for φ(·, p) ∈ L1 (Ω). We also
refer the reader to [15] for lower semicontinuity and approximation theorems of functionals∫
Ω f(x,Du), u ∈ BV (Ω) , using the work of Reshetnyak; and, for example, in [1] for the
relaxation in BV (Ω) with respect to the L1 norm for functionals

∫
Ω f(x, u,∇u) dx defined

on W 1,1
(
Ω;Sd−1

)
for Ω ⊂ RN open and bounded and Sd−1 the unit sphere in Rd. How-

ever the integrands f(x, p) and f(x, z, p) are always assumed to be lower semicontinuous
or continuous on Ω× RN or Ω× R× RN respectively for these cases.

Importantly, we note that the approximation Lemma 6.2 in [2] is used to prove existence
results there for the solution to the time dependent problem

∂u
∂t = div∇pg(x,Du) in (0,∞)× Ω
u(t, x) = h(x) on (0,∞)× ∂Ω
u(0, x) = u0(x) for x ∈ Ω

via the strong solution using the theory of semigroups in L2, which corresponds to the
stationary problem

min
u∈BV (Ω)∩L2(Ω)

Φφ(u),

with

Φφ(u) :=

∫
Ω
g(x,Du) +

∫
∂Φ

|h− u|g0(x, ν(x)) dHN−1,

for given boundary data h. Here g is continuous on Ω × RN , convex and continuously
differentiable in the second variable p, and

g0(x, p) := lim
t→0+

tg(x, p/t).

Appropriately defined solutions of the above time flow in L1 using similar semigroup
methods are also proved there.

2. Main Results

As stated in the Introduction, we prove an approximation result for a class of function-
als
∫
Ω φ(x,Du) by

∫
Ω φ(x,∇uk), uk ∈ W 1,1 (Ω) ∩ C∞ (Ω) where φ(x, p) satisfies (1)-(3)



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2028

and with an additional structure condition on g. Here we will use, from [6], the conjugate
function g∗ for given g:

g∗(x, q) := sup
p∈RN

{q · p− g(x, p)}.

If g is convex in p, then it is easy to show that g∗ is convex in q. Also if g is additionally
continuous in p, then for a.e. x, there holds g(x, p) = g∗∗(x, p) for all p ∈ RN (see [6],[4]).

We will need the following lemma which is Proposition 1, from [17], which for the
convenience of the reader we restate here.

In the sequel we define

V :=
{
ϕ ∈ C1

0 (Ω,RN ) : |ϕ(x)| ≤ ψ(x) for all x ∈ Ω
}
.

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) ∩ L∞ (Ω) , ψ ≥ 0, k(x, u) ∈ 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| (3)

= sup
ϕ∈V

{
−
∫
Ω
udivϕ+ φ∗(x, ϕ(x)) dx

}
.

If in addition ∂Ω is Lipschitz, u ∈ BV (Ω) , then we have the continuous trace operator
T : BV (Ω) → L1(∂Ω,HN−1) ([7]). Thus if h ∈ BV (Ω),

Gh(u) =
∫
Ω
φ(x,∇u) dx+

∫
Ω
ψ(x)|Dsu|+

∫
∂Ω

|u− h| dHN−1 (4)

= sup
{ϕ∈C1(Ω,RN):|ϕ|≤ψ(x)}

{
−
∫
Ω
udivϕ+ φ∗(x, ϕ(x)) dx+

∫
∂Ω
ϕn̂h dHN−1

}
.

Furthermore, both G and Gh are lower semicontinuous in L1.



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2029

Before we state the proof, we note that the lower semicontinuity of G and Gh in L1 is
not covered the results in [10], [11], [13] since we only assume φ(·, p) ∈ L1 (Ω) for each p
and hence the condition that

lim
x̃→x,t→∞

tφ(x̃, p/t) exists

as stated there may not hold if φ(·, p) is only assumed to be in L1 (Ω). Also see [18] for
more general results for lower semicontinuity.

For an example of an integrand φ satisfying the conditions of Lemma 1, consider the
following G with α ∈ L1 (Ω) , δ > 0:

G(u) :=
∫
Ω
φ(x,Du)

with

φ(x, p) =

{
ψ(x)

√
α2(x) + δ + |p|2 if |p| ≤ β

ψ(x)|p|+ ψ(x) α(x)+δ√
α2(x)+δ+β2+β

if |p| > β.

We now state the approximation theorem.

Theorem 1. Let G and Gh be as defined in Lemma 1 with φ satisfying the same conditions.
Then for each u ∈ BV (Ω) ∩ Lr (Ω) , 1 ≤ r < ∞ there exist a sequence uk ∈ W 1,1 (Ω) ∩
C∞ (Ω) ∩ Lr (Ω) with

G(uk) → G(u) and
uk → u in Lr (Ω) .

In addition, if ∂Ω is Lipschitz and h ∈ L1(∂Ω) we have for each u ∈ BV (Ω) a sequence
uk ∈W 1,1 (Ω) ∩ C∞ (Ω) ∩ Lr (Ω) with

Gh(uk) → Gh(u),
uk → u in Lr (Ω) , and

Tuk = Tu

where Tw is the trace operator for w ∈ BV (Ω) .

Proof. We follow [8] taking into account the extra φ∗ term.
Fix ε > 0 and construct an open covering {Ai} of Ω where Ai = Ωi+1−Ωi−1, A1 = Ω2

where
Ωk = {x ∈ Ω : dist(x, ∂Ω) > 1/(m+ k)} , k = 0, 1, 2, ...

and with m so large that ∫
Ω−Ω0

ψ(x)|Du| < ε and (5)

|Ω− Ω1| ≤ ε (6)



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2030

Now construct a sequence {uε} so that

uε =

∞∑
i=1

ηεi ∗ (uϕi)

where η is the usual mollifier on RN , {ϕi} is a partition of unity subordinate to {Ai} , and
the εi are chosen to that the four conditions all hold:

1. each εi < ε, i ≥ 1
2.
∫
Ω |ηεi ∗ (uϕi)− uϕi|r dx ≤ ε2−i

3.
∫
Ω |ηεi ∗ (u∇ϕi)− u∇ϕi| dx ≤ ε2−i

4. support ηεi ∗ (uϕi) ⊂ Ωi+2 − Ωi−2.
Summing over all i gives∫

Ω
|uε − u| dx ≤

∞∑
i=1

∫
Ω
|ηεi ∗ (uϕi)− uϕi| dx ≤ ε

giving uε → u in L1 (Ω) . Hence by L1 lower semicontinuity in Lemma 1∫
Ω
φ(x,Du) ≤ lim inf

ε→0

∫
Ω
φ(x,Duε). (7)

First we note that |(ϕ1ηε1 ∗ ϕ)(x)| ≤ ψ(x) + ω(ε1) where the modulus of continuity ω of
ψ satisfies ω(ε1) → 0 as ε1 → 0, and that for φc(x, p) := φ(x, p) + c|p|, for each c > 0,
satisfies the same assumptions on φ. Hence for each u ∈ BV (Ω)

sup
|ϕ(x)|≤ψ(x)+c

{
−
∫
Ω
udivϕ+ φ∗

c(x, ϕ(x)) dx

}
(8)

=

∫
Ω
φ(x,∇u) + c|∇u| dx+

∫
Ω
(ψ(x) + c )d|Dsu|.

Now let ϕ ∈ C1
0 (Ω;RN ) with |ϕ(x)| ≤ ψ(x) each x, then

−
∫
Ω
uεdivϕ+ φ∗

ω(ε1)
(x, ϕ(x)) dx =

( ∞∑
i=1

−
∫
Ω
(ηεi ∗ (uϕi))divϕ dx

)
(9)

−
∫
Ω
φ∗
ω(ε1)

(x, ϕ(x)) dx (10)

= −
∫
Ω
udiv(ϕ1ηε1 ∗ ϕ) dx−

∫
Ω
φ∗
ω(ε1)

(x, ϕ(x)) dx−
∞∑
i=2

∫
Ω
udiv(ϕiηεi ∗ ϕ) dx

+
∞∑
i=1

∫
Ω
ϕ(ηεi ∗ (u∇ϕi)− u∇ϕi) dx

= −
∫
Ω
udiv(ϕ1ηε1 ∗ ϕ) + φ∗

ω(ε1)
(x, ηε1 ∗ ϕ) dx−

∞∑
i=2

∫
Ω
udiv(ϕiηεi ∗ ϕ) dx



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2031

+

∞∑
i=1

∫
Ω
ϕ(ηεi ∗ (u∇ϕi)− u∇ϕi) dx

+

∫
Ω
φ∗
ω(ε1)

(x, ηε1 ∗ ϕ)− φ∗
ω(ε1)

(x, ϕ(x)) dx := I + II + III + IV.

By Lemma 3 in [16] we have from the Lipschitz property of φ∗
ω(ε1)

IV ≤
∫
Ω
|φ∗
ω(ε1)

(x, ηε1 ∗ ϕ)− φ∗
ω(ε1)

(x, ϕ(x))| dx

≤ β

∫
Ω
|ηε1 ∗ ϕ− ϕ| dx.

We now in addition to 1-4 choose ε1 so that
∫
Ω1

|ηε1 ∗ ϕ− ϕ| dx ≤ ε. The since |ηε1 ∗ ϕ| ≤
∥ψ∥∞ we then have

IV ≤ β

∫
Ω
|ηε1 ∗ ϕ− ϕ| dx

= β

∫
Ω1

|ηε1 ∗ ϕ− ϕ| dx+ β

∫
Ω−Ω1

|ηε1 ∗ ϕ− ϕ| dx

≤ βε+ 2β ∥ψ∥ ε→ 0 as ε→ 0.

Also, we have as in [8]
III, II → 0 as ε→ 0.

Now

I = −
∫
Ω
udiv(ϕ1ηε1 ∗ ϕ) + φ∗

ω(ε1)
(x, ηε1 ∗ ϕ) dx

= −
∫
Ω
udiv(ϕ1ηε1 ∗ ϕ) + φ∗

ω(ε1)
(x, ϕ1ηε1 ∗ ϕ) dx

+

∫
Ω
φ∗
ω(ε1)

(x, ϕ1ηε1 ∗ ϕ)− φ∗
ω(ε1)

(x, ηε1 ∗ ϕ) dx.

Again from Lemma 3 in [16] we have for the last line

|η| : =

∣∣∣∣∫
Ω
φ∗
ω(ε1)

(x, ϕ1ηε1 ∗ ϕ)− φ∗
ω(ε1)

(x, ηε1 ∗ ϕ) dx
∣∣∣∣

≤ β

∫
Ω
|ϕ1ηε1 ∗ ϕ− ηε1 ∗ ϕ| dx

= β

∫
Ω−Ω1

|ϕ1 − 1||ηε1 ∗ ϕ| dx

≤ 2β

∫
Ω−Ω1

|ηε1 ∗ ϕ| dx

≤ 2β ∥ψ∥∞ ε



T. Wunderli / Eur. J. Pure Appl. Math, 16 (4) (2023), 2025-2034 2032

since ϕ1 ≡ 1 on Ω1.
Therefore η → 0 as ε→ 0.
Thus from (8)

I = −
∫
Ω
udiv(ϕ1ηε1 ∗ ϕ) + φ∗

ω(ε1)
(x, ϕ1ηε1 ∗ ϕ) dx+ η

≤
∫
Ω
φ(x,∇u) + ω(ε1)|∇u| dx+

∫
Ω
(ψ(x) + ω(ε1) ) d|Dsu|+ η

=

∫
Ω
φ(x,Du) +

∫
Ω
ω(ε1)|∇u| dx+ ω(ε1)

∫
Ω
d|Dsu|+ η,

keeping in mind that the last three terms approach 0 as ε → 0. Thus we have from (9)
and for each ϕ with |ϕ(x)| ≤ ψ(x),

−
∫
Ω
uεdivϕ+ φ∗(x, ϕ(x)) dx

≤ I + II + III + IV +

∫
Ω
|φ∗(x, ϕ(x)) dx− φ∗

ω(ε1)
(x, ϕ(x))| dx

= I + II + III + IV +

∫
Ω
|φ∗(x, ϕ(x))− (φ(x, ϕ(x)) + ω(ε1)|ϕ(x)|)∗dx

≤ I + II + III + IV + ω(ε1)|ψ|∞ |Ω|

≤
∫
Ω
φ(x,Du) +

∫
Ω
ω(ε1)|∇u| dx+ ω(ε1)

∫
Ω
d|Dsu|+ η

+II + III + IV + ω(ε1)|ψ|∞ |Ω| .

The second inequality follows from the note before (8), the assumption |ϕ(x)| ≤ ψ(x), and
Lemma 2 in [16]. Thus we have

−
∫
Ω
uεdivϕ+ φ∗(x, ϕ(x)) dx ≤

∫
Ω
φ(x,Du) +

∫
Ω
ω(ε1)|∇u| dx

+ω(ε1)

∫
Ω
d|Dsu|+ η (11)

+II + III + IV + ω(ε1)|ψ|∞ |Ω| .

Taking the supremum over all such ϕ with |ϕ(x)| ≤ ψ(x) in (11), and then letting ε → 0
we have

lim sup
ε→0

−
∫
Ω
φ(x,Duε)dx ≤

∫
Ω
φ(x,Du).

Combining with (7) gives the result. The second part of the theorem is proved as in the
first case and as in [5] for the boundary term.

Combining Lemma 1 and Theorem 1 we have the following extension of Theorem 6.4
in [2].



REFERENCES 2033

Theorem 2. Let φ satisfy the conditions of Lemma 1 and Theorem 1, then

inf
u∈BV (Ω)

G(u) = inf

{∫
Ω
φ(x,∇u) dx : u ∈W 1,1(Ω)

}
, and

inf
u∈BV (Ω), u=h on ∂Ω

Gh(u) = inf

{∫
Ω
φ(x,∇u) dx : u ∈W 1,1 (Ω) and u = h on ∂Ω

}
.

In addition, G, Gh is the greatest L1 (Ω)-lower semicontinuous functional on BV (Ω) sat-
isfying G(u) ≤

∫
Ω φ(x,∇u) dx, and Gh(u) ≤

∫
Ω φ(x,∇u) dx for all u ∈ W 1,1(Ω) and

u ∈W 1,1(Ω) with u = h on ∂Ω respectively.

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