# Lower Semicontinuity inL1 of a Class of Functionals Defined on BV with Carathéodory Integrands

Authors: T. Wunderli
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

We prove lower semicontinuity inL1Ω for a class of functionals G:BVΩ⟶ℝ of the form Gu=∫Ωgx,∇udx+∫ΩψxdDsu where g:Ω×ℝN⟶ℝ, Ω⊂ℝN is open and bounded, g·,p∈L1Ω for each p, satisfies the linear growth condition limp⟶∞gx,p/p=ψx∈CΩ∩L∞Ω, and is convex in p depending only on p for a.e. x. Here, we recall for u∈BVΩ; the gradient measure Du=∇udx+dDsux is decomposed into mutually singular measures ∇udx and dDsux. As an example, we use this to prove that ∫Ωψxα2x+∇u2dx+∫ΩψxdDsu is lower semicontinuous in L1Ω for any bounded continuous ψ and any α∈L1Ω. Under minor addtional assumptions on g, we then have the existence of minimizers of functionals to variational problems of the form Gu+u−u0L1 for the given u0∈L1Ω, due to the compactness of BVΩ in L1Ω.


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## Body

## 1. Introduction

We prove anL1 lower semicontinuity result for convex linear growth functionals ∫Ωgx,Du, defined on BV, whose integrands g:Ω×ℝN⟶ℝ are radially symmetric in p for a.e. x∈Ω, that is, gx,p depends on x,∣p∣,g·,p∈L1Ω for each p, and satisfies a fairly general structure condition. Our results expand the class of integrands from those of the form
(1)φx,p=gx,p,ifp≤β,ψxp+kx,ifp>β,as presented in [1], for which lower semicontinuity in L1 holds.We use the conjugate functiong∗ of g to prove our main result, Theorem 1 in Section 3. The conjugate function is used, for example, in [2] to approximate ∫Ωgx,Du for the given u∈BVΩ by a sequence ∫Ωgx,∇un for un∈W1,1Ω to prove the existence of the corresponding gradient time flow, although g is assumed to be continuous in x in these cases. Thus, one advantage of Theorem 1 in this paper is that we can also obtain the existence results for time flow by deriving a similar convergence result for our case, but with no continuity assumption in the x variable. In fact for the results presented here, g may contain singularities in x, as in Example 3. In addition, the corresponding convergence results in [3] assume another continuity condition on g in x, similar to (3) which is not covered by our assumptions on g.We note that the integrands considered in this paper (of the form (1)) have been used in models in applications of image processing [2]. However, as mentioned above, the main result of this paper covers a larger class of integrands.We assume throughout, unless otherwise stated, thatΩ⊂ℝN is bounded and open and g is radially symmetric in p for a.e. x∈Ω and g is convex in p for a.e. x∈Ω so thatgx,λ1p1+λ2p2≤λ1gx,p1+λ2gx,p2 for each p1,p2∈ℝN,0≤λ1,λ2≤1,λ1+λ2=1, and g∈L1Ω for each p∈ℝN. Since for a.e. x,g is convex and real valued in p, it is well known that g must be continuous in p for a.e. x; hence, g is a Carathéodory function. Furthermore, we assume the linear growth of g so that
(2)limp⟶∞gx,pp=ψx,fora.e.x∈Ω,where ψ∈CΩ∩L∞Ω,ψ≥0.As stated above, we make no continuity assumption for thex variable for g. Additionally, in contrast to the works of [3–8] and [9] we do not assume g to be lower semicontinuous in x,p. Also, our assumptions on g are not covered by the class of integrands EΩ;ℝN and RΩ;ℝN in [10, 11]. The integrand class E in [10] requires joint continuity of g in x and p, and R and the integrands in [11] require g for our case be defined on Ω¯×ℝN with
(3)limt⟶∞,x′⟶xgx′,tpt=ψx,foreachx∈Ω¯,p∈ℝN,which may not hold if it is only assumed g∈L1Ω for each p, as g may contain singularities.


## 2. Mathematical Preliminaries

We recall by definition thatu∈BVΩ if and only if u∈L1Ω and
(4)∫ΩDu≔supϕ∈C0∞Ω,ℝN,ϕx≤1allx∈Ω−∫Ωudivϕdx<∞,in which case the total variation measure Du is decomposed in to Du=∇udx+dDsux where ∇udx<<LN and Dsu⊥LN using the Lebesgue decomposition theorem [12]. Functionals defined for u∈BVΩ with Carathéodory integrands gx,p of linear growth (2) and the convex in the p variable are defined [4, 5, 13, 14] by
(5)∫Ωgx,Du=∫Ωgx,∇udx+∫ΩψxdDsux.However, it is not immediate that functionals∫Ωgx,Du defined by (5) are lower semicontinuous in L1Ω. As noted above, lower semicontinuity was was proven for certain integrands g, but to the best of our knowledge, there is no general L1 lower semicontinuity result for convex Carathéodory functions g where for each p, g·,p∈L1Ω.We will also use the conjugate functiong∗ of g where g∗x,q≔supp∈ℝNq·p−gx,p [8]. We note that as g is convex in p and g∗ is convex in q.In [1], L1 lower semicontinuity of ∫Ωφx,Du for integrands of the form (1) is proved for φ:Ω×ℝN⟶ℝ convex in p,φx,p radially symmetric in p for a.e. x∈Ω,φ·,p∈L1Ω and a fairly general structure condition on φ which does not assume continuity in x. The proof is based on proving that
(6)∫Ωφx,∇udx+∫ΩψxdDsu=supϕ∈C01Ω,ℝN:ϕx≤ψxforallx∈Ω∫Ω∇u·ϕx−φ∗x,ϕxdx=supϕ∈C01Ω,ℝN:ϕx≤ψxforallx∈Ω−∫Ωudivϕx+φ∗x,ϕxdx,where φ∗ is the conjugate function of φ and the last equality follows from integration by parts for u∈BVΩ [12]. Lower semicontinuity in L1Ω immediately follows as the final equality is the supremum of functionals, each L1 continuous in u. In the next section, we use the method above to prove our main result, Theorem 1.


## 3. Main Results

We first define(7)V=ϕ∈C01Ω,ℝN:ϕx≤ψx,forallx∈Ω.Theorem 1.
Assumeg:Ω×ℝN⟶ℝ with
(8)=ψx∈CΩ∩L∞Ω.gx,p are both radially symmetric and convex in p for a.e. x, and if for each M>0,
(9)φMx,p≔gx,p,ifp≤M,ψxp+gx,M−ψxM,ifp>M,is convex in p and there exists fM∈L1Ω such that gx,p−φMx,p≤fMx a.e. x, for all p≥M, where ∫ΩfMxdx⟶0 as M⟶∞. Additionally, assume the following structure condition on g: that is, for some G, we have gx,p=Gr1x,⋯,rkx,pforallp where Gz1,⋯,zK,p=gz1,⋯,zk,p and where g is C1 in the variable z=z1⋯,zK∈U⊂ℝK,U open,ri∈L1Ω each i,r1x,⋯,rKx∈U a.e. x, and ∇zgz,p≤C,C independent of z,p, p≤M for each M. Then, for ∫Ωgx,Du defined by
(10)∫Ωgx,Du≔supV∫Ω∇u·ϕx−g∗x,ϕxdx,we have in fact
(11)∫Ωgx,Du=∫Ωgx,∇udx+∫ΩψxdDsu.Thus, the functional(12)Gu=∫Ωgx,∇udx+∫ΩψxdDsu,defined on BVΩ, is lower semicontinuous in L1Ω, that is, if un⟶u in L1Ω, then, Gu≤liminfn⟶∞Gun. Moreover, if ∂Ω is Lipschitz, then, for the given h∈L1∂Ω,
(13)∫Ωgx,∇u+∫ΩψxdDsux+∫∂Ωψxu−hdHN−1=supϕ∈C1Ω¯,ℝN:∣ϕ∣≤ψx−∫Ωudivϕ+g∗x,ϕxdx+∫∂Ωϕ·n^hdHN−1,and hence, the functional
(14)Ghu=∫Ωgx,∇u+∫ΩψxdDsux+∫∂Ωψxu−hdHN−1,defined on BVΩ, is lower semicontinuous in L1Ω. Here, u is defined on ∂Ω in the sense of trace [12].Proof.
From the above assumptions ong, we have
(15)g∗x,q=supp∈ℝNp·q−gx,p≤supp∈ℝNp·q−φMx,p+supp∈ℝNφMx,p−gx,p=supp∈ℝNp·q−φMx,p+supp≥MφMx,p−gx,p≤φM∗x,q+fMx.
Similarly, we haveφM∗x,q≤g∗x,q+fMx givingg∗x,q−φM∗x,q≤fMxforallq≤ψx.
From the above estimate forg∗−φ∗, we have g∗x,q=∞ if and only if φM∗x,q=∞ if and only if q≤ψx, and hence,
(16)gx,p=supq≤ψxp·q−g∗x,q.☐Now,(17)∫Ωgx,Du=supV∫Ω∇u·ϕx−g∗x,ϕxdx=supV∫Ω∇u·ϕx−φM∗x,ϕxdx+supV∫Ω∇u·ϕx−φM∗x,ϕx+ε1x,ϕxdx−supV∫Ω∇u·ϕx−φM∗x,ϕxdx=∫ΩφMx,∇udx+∫ΩψxdDsu+ε2,with ε1x,ϕx≔φM∗x,ϕx−g∗x,ϕx and ε1x,ϕx≤fMx from the above and ε2 is defined by
(18)ε2≔supV∫Ω∇u·ϕx−φM∗x,ϕx+ε1x,ϕxdx−supV∫Ω∇u·ϕx−φM∗x,ϕxdx.We now showε2⟶0 as M⟶∞. In fact,
(19)∫Ω∇u·ϕx−φM∗x,ϕx+ε1x,ϕxdx≤∫Ω∇u·ϕx−φM∗x,ϕxdx+∫ΩfMxdx,from ε1x,ϕx≤fMx. Taking supremum over both sides gives
(20)ε2=supV∫Ω∇u·ϕx−φM∗x,ϕx+ε1x,ϕxdx−supV∫Ω∇u·ϕx−φM∗x,ϕxdx≤∫ΩfMxdx.Similarly, we have−ε2≤∫MfMxdx giving
(21)ε2≤∫ΩfMxdx⟶0asM⟶∞.Now, letM⟶∞ in (17) to get
(22)∫Ωgx,Du=∫Ωgx,∇udx+∫ΩψxdDsu.For the second claim in the theorem, as∂Ω is Lipschitz, the continuous trace operator [12])T:BVΩ⟶L1∂Ω,HN−1 exists and (13) follows as in the proof above. Finally, lower semicontinuity of Gh follows from Theorem 5 in [1].Remark 2.
We note that the condition∇zgz,p≤C may be modified if in the expression gz1,⋯,zK,p, one of the zi′s corresponds to ψx. That is, if, e.g., zK=ψx, we may only require that ∇z1,⋯,zK−1gz,p≤C as each x∈Ω is a Lebesgue point of ψ by continuity. In fact, we have, noting that g∗x,q=g∗r1x,⋯,rK−1x,ψx,q,(23)1Bρ∫Bρxg∗r1x,⋯,rK−1x,ψx,p−g∗r1y,⋯,rK−1y,ψy,pdy≤1Bρ∫Bρxg∗r1x,⋯,rK−1x,ψx,p−g∗r1x,⋯,rK−1x,ψy,pdy+1Bρ∫Bρxg∗r1x,⋯,rK−1x,ψy,p−g∗r1y,⋯,rK−1y,ψy,pdy≤1Bρ∫Bρxg∗r1x,⋯,rK−1x,ψx,p−g∗r1x,⋯,rK−1x,ψy,pdy+1Bρ∫Bρxsupz,p∇z1,⋯,zK−1g∗z,p·r1x,⋯,rK−1x−r1y,⋯,rK−1ydy.The last term is bounded by1/∣Bρ∣1/∣Bρ∣∫BρxCr1x,⋯,rK−1x−r1y,⋯,rK−1ydx which approaches 0 as ρ⟶0 on the common Lebesgue set of r1,⋯,rk−1. The next to last term approaches 0 a.e. x as ρ⟶0 since g∗r1x,⋯,rK−1x,ψy,p is continuous in y wherever r1x,⋯,rK−1x are defined. The Lebesgue set of g∗ thus contains the Lebesgue set of r1,⋯,rk independent of p. The rest follows exactly as in the proof of Theorem 4 in [15], but with ∇z1,⋯,zK−1Gεz,p and ∇z1,⋯,zK−1Gε∗z,p replacing ∇zGεz,p and ∇zGε∗z,p and Remark 2 in [1].Example 3.
Forα∈L1Ω, u∈BVΩ,ψ∈CΩ∩L∞Ω, the functional,
(24)Fu=∫Ωψxα2x+∇u2dx+∫ΩψxdDsu,is lower semicontinuous on L1Ω.Proof.
Lettinggδx,p=ψxα2x+δ+p2,α∈L1Ω,δ>0 we have
(25)φMx,p=ψxα2x+δ+p2,ifp≤M,ψxp+kMx,ifp>M,where
(26)kMx=ψxαx+δα2x+δ+M2+M.Letting αδ=α+δ, we have for p≥M(27)gδx,p−φMx,p=ψxαxδα2xδ+p2+p−αxδα2xδ+M2+M≤2ψxαxδα2xδ+M2+M≔fMx⟶0asM⟶∞,and hence, ∫ΩfMdx⟶0 as M⟶∞ by Lebesgue’s dominated convergence theorem, as fMx≤2ψx a.e. Note that φM is convex in t since by defining φ~M:Ω×0,∞⟶ℝ(28)φ~Mx,t=ψxα2x+δ+t2,if0≤t≤M,ψxt+kMx,ift>M.
We see the left derivative∂/∂tφ~Mx,t at t=M is
(29)φ~M′x,M−=ψxtαδ2x+t2,while the right derivative at t=M is φ~M′x,M+=ψx≥ψxt/αδ2x+t2=φ~M′x,M−. Thus, φ~M is convex in t. As φ~M is also increasing in t, we have that φMx,p=φ~Mx,p is convex in p.☐The conditions for Theorem1 are thus satisfied for gδ and φM with
(30)Gz1,z2,p=gδz1,z2,p=z2z12+δ+p2,U=ℝ2,∂gδ∂z1z1,z2,p=z2z1z12+δ+p2≤z2≤ψ,forallz1,z2,p∈ℝ2×ℝN,noting Remark 2. Hence,
(31)∫Ωgδx,Du=∫Ωgδx,∇udx+∫ΩψxdDsu.For the casegx,p=ψxα2x+p2,α∈L1Ω, we note that
(32)gδx,p−gx,p≤ψxδα2x+δ+p2+α2x+p2≤δ,for a.e. x∈Ω and for each p∈ℝN. As in the proof of Lemma 2 in [16] and Theorem 1 above, we have for a.e x, all q∈ℝN,(33)g∗x,q−gδ∗x,q≤gδx,p−gx,p≤δ,and similar to the above estimates (17), for each δ>0, we have
(34)∫Ωgx,Du−∫Ωgδx,∇udx+∫ΩψxdDsu≤∫Ωδdx.Lettingδ⟶0 gives
(35)∫Ωgx,Du≔supV∫Ω∇u·ϕx−g∗x,ϕxdx=∫Ωgx,∇udx+∫ΩψxdDsu.Lower semicontinuity ofF immediately follows.We finally note that a version of Theorem1, along with Remark 2, holds for nonradially symmetric integrands g, but with the additional smoothness assumption that for a.e. x∈Ω,gx,·∈C2ℝN.Theorem 4.
Assume thatg:Ω×ℝN⟶ℝ with
(36)limp⟶∞gx,pp=ψx∈CΩ∩L∞Ω,wheregx,p is convex and C2 in p for a.e. x, and if for each M>0,
(37)φMx,p≔gx,p,ifp≤M,ψxp+gx,M−ψxM,ifp>M,is both convex and C1 in p and there exists fM∈L1Ω such that gx,p−φMx,p≤fMx a.e. x, for all p≥M, where ∫ΩfMxdx⟶0 as M⟶∞. Additionally assume the following structure condition on g: that is, for some G we have gx,p=Gr1x,⋯,rkx,pforallp whereGz1,⋯,zK,p=gz1,⋯,zk,p and where g is C1 in the variable z=z1⋯,zK∈U⊂ℝK,U open,ri∈L1Ω each i,r1x,⋯,rKx∈U a.e. x, and ∣∇zgz,p∣≤C,C independent of z,p, ∣p∣≤M for each M. Then
(38)∫Ωgx,Du≔supV∫Ω∇u·ϕx−g∗x,ϕxdx=∫Ωgx,∇udx+∫ΩψxdDsu.Thus, the functional(39)Gu=∫Ωgx,∇udx+∫ΩψxdDsu,defined on BVΩ, is lower semicontinuous in L1Ω. Moreover, if ∂Ω is Lipschitz, then, for the given h∈L1∂Ω,
(40)∫Ωgx,∇u+∫ΩψxdDsux+∫∂Ωψxu−hdHN−1=supϕ∈C1Ω¯,ℝN:∣ϕ∣≤ψx−∫Ωudivϕ+g∗x,ϕxdx+∫∂Ωϕ·n^hdHN−1,and hence, the functional
(41)Ghu=∫Ωgx,∇u+∫ΩψxdDsux+∫∂Ωψxu−hdHN−1,defined on BVΩ, is lower semicontinuous in L1Ω.Proof.
The proof is the same as the proof of Theorem1, noting Theorem 4 in [15] and Remark 2 in [1].☐We immediately have from standard theory the following existence result:(42)infΩψx=c1>0,(43)gx,p≥c2p,forsomec2>0.Corollary 5.
Letg satisfy the assumptions of Theorems 1 and 4 or Remark 2. If in addition we haveThen, for the givenu0∈L1Ω, the functionals
(44)Φu=∫Ωgx,Du+u−u0L1Ω,Φhu=∫Ωgx,Du+∫∂Ωψxu−hdHN−1+u−u0L1Ω,have a minimizer in BVΩ. Furthermore, the minimizer is unique if g is strictly convex in p.Proof.
ForΦ, this follows from lower semicontinuity of Φ in L1 and standard compactness results for BV, noting that assumptions (42) and (43) imply using (5) that
(45)∫ΩDu≤minc1,c2∫Ωgx,Du.
Thus, minimizing sequencesun of Φ are bounded in the BV norm uBVΩ≔∫ΩDu+uL1Ω so that there is u∈L1Ω with un⟶u in L1Ω [12], and hence,
(46)Φu≤liminfn⟶∞Φun=minv∈L1ΩΦv.
The proof is essentially the same forΦh using (13). Finally, if g is strictly convex, then, so is Φ and Φh. Thus, if there are minimizers u1≠u2, then, we have
(47)Φu1+u22<minv∈L1ΩΦv,Φhu1+u22<minv∈L1ΩΦhv,
a contradiction.☐We finally remark, as noted in [1], that Theorems 1 and 4 of this paper may be extended to vector-valued functions ux=u1x,⋯,uMx where Du is an M×N matrix with Dui∈BVΩ for each i and ∫Ωgx,Du is defined by writing Du as a vector of length NM with g:Ω×ℝNM⟶ℝ and g depending on x,Du for the case of Theorem 1. We may also consider integrands ∫Ωgx,u,Du with appropriate assumptions on gx,z,p, such as Lipschitz continuity in z, using similar methods as presented here and in [1, 15, 16].


## 4. Conclusion

In this paper, we have expanded the class of functionals∫Ωgx,Du defined on the BV space which are L1Ω lower semicontinuous to include certain integrands gx,p which, for each p∈ℝN, are only assumed to be in L1Ω. The structure condition for which lower semicontinuity holds is fairly general and is for many cases not difficult to verify. Furthermore, as mentioned above, using the method presented here, we may expand the main theorem of this paper to include functionals of the form ∫Ωgx,u,Du for vector-valued functions u:Ω⟶ℝM. Finally, as noted above, lower semicontinuity is essential to proving the existence of minimizers of functionals of the form Φu=∫Ωgx,Du+u−u0L1Ω over BVΩ.



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