# A New Extension of Serrin's Lower Semicontinuity Theorem

Authors: Xiaohong Hu; Shiqing Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional∫Ωfx,u,u′dx defined on Wloc1,1Ω is lower semicontinuous with respect to the strong convergence in Lloc1, under the assumptions that the integrand fx,s,ξ has the locally absolute continuity about the variable x.


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## Body

## 1. Introduction and Main Results

The aim of this paper is to give some new sufficient conditions for lower semicontinuity with respect to the strong convergence inLloc1 for integral functionals
(1)F(u,Ω)=∫Ω‍f(x,u(x),Du(x))dx,
where Ω is an open set of Rn, u∈Wloc1,1(Ω), defined on Wloc1,1(Ω)={u:u∈L1(K),Du∈L1(K),forallK⊂⊂Ω} [1], Du denotes the generalized gradient of u, and the integrand f(x,s,ξ):Ω×R×Rn→[0,∞) satisfies the following condition:(H1)
f is continuous in Ω×R×Rn, and f(x,s,ξ) is convex in ξ∈Rn for any fixed (x,s)∈Ω×R.The integral functionalF is called lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1, if, for every um,u∈Wloc1,1(Ω), such that um→u in Lloc1, then
(2)liminfm→+∞F(um,Ω)≥F(u,Ω).
It is well known that condition (H1) is not sufficient for lower semicontinuity of the integral F in (1) (see book [2]). In addition to (H1), Serrin [3] proposed some sufficient conditions for lower semicontinuity of the integral F. One of the most known conclusions is the following one.Theorem 1 (see [3]).
In addition to (H1), if f satisfies one of the following conditions:(a)
f(x,s,ξ)→+∞ when |ξ|→+∞, for all (x,s)∈Ω×R,(b)
f(x,s,ξ) is strictly convex in ξ∈Rn for all (x,s)∈Ω×R,(c)
the derivativesfx(x,s,ξ), fξ(x,s,ξ), and fξx(x,s,ξ) exist and are continuous for all (x,s,ξ)∈Ω×R×Rn.thenF(u,Ω) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1.Conditions (a), (b), and (c) quoted above are clearly independent, in the sense that we can find a continuous functionf satisfying just one of them but none of the others. Many scholars have weakened the conditions of integrand f and generalized Theorem 1, such as Ambrosio et al. [4], Cicco and Leoni [5], Fonseca and Leoni [6, 7]. In particular Gori et al. [8, 9] proved the following theorems.Theorem 2 (see [8, 9]).
Let one assume thatf satisfies (H1) and that, for every compact set K⊂Ω×R×Rn, there exists a constant L=L(K) such that
(3)|fξ(x1,s,ξ)-fξ(x2,s,ξ)|≤L|x1-x2|,∀(x1,s,ξ),(x2,s,ξ)∈K,
and, for every compact set K1⊂Ω×R, there exists a constant L1=L1(K1) such that
(4)|fξ(x,s,ξ)|≤L1,∀(x,s)∈K1,∀ξ∈Rn,|fξ(x,s,ξ1)-fξ(x,s,ξ2)|≤L1|ξ1-ξ2|,∀(x,s)∈K1,∀ξ1,ξ2∈Rn.
Then F(u,Ω) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1.Theorem 3 (see [8, 9]).
Let f satisfy (H1) such that, for every open setΩ′×H×K⊂⊂Ω×R×Rn, there exists a constant L=LΩ′×H×K such that, for every x1,x2∈Ω′, s∈H, and ξ∈K,
(5)|f(x1,s,ξ)-f(x2,s,ξ)|≤L|x1-x2|.
Then the functional F(u,Ω) is lower semicontinuous in Wloc1,1(Ω) with respect to the Lloc1 convergence.Condition (5) means that f is locally Lipschitz continuous with respect to x, that is, the Lipschitz constant is not uniform for (s,ξ)∈R×Rn. This is an improvement of (c) of Serrin’s Theorem 1. Then a question arises, that is whether there are weaker enough conditions more than locally Lipschitz continuity. In this paper, we consider absolute continuity. Obviously, absolute continuity is weaker than Lipschitz continuity. The following theorems show that, in addition to (H1), the locally absolute continuity on f about x is sufficient for the lower semicontinuity of the variational functional.Theorem 4.
LetΩ⊂R be an open set; f(x,s,ξ):Ω×R×R→[0,+∞) satisfies the following conditions:(H1)
f(x,s,ξ) is continuous on Ω×R×R, and, f(x,s,ξ) is convex in ξ∈R for all (x,s)∈Ω×R;(H2)
fξ(x,s,ξ) is continuous on Ω×R×R, and for every compact set of Ω×R×R, fξ(x,s,ξ) is absolutely continuous about x;(H3)
for every compact setK1⊂Ω×R, there exists a constant L1=L1(K1), such that
(6)|fξ|≤L1,∀(x,s)∈K1,∀ξ∈R,(7)|fξ(x,s,ξ1)-fξ(x,s,ξ2)|≤L1|ξ1-ξ2|,∀(x,s)∈K1,∀ξ1,ξ2∈R. Then the functional F(u,Ω)=∫Ω‍f(x,u(x),u′(x))dx is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1(Ω).Theorem 5.
LetΩ⊂R be an open set; f(x,s,ξ):Ω×R×R→[0,+∞) satisfies (H1) and the following condition:(H4)
for every compact set ofΩ×R×R, f(x,s,ξ) is absolutely continuous about x.Then the functional F(u,Ω) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1(Ω).


## 2. Preliminaries

In this section, we will collect some basic facts which will be used in the proofs of Theorems4 and 5.It is well know that a real functionf:[a,b]→R is called an absolutely continuous function on [a,b], if, forallε>0, ∃δ>0, such that for any finite disjoint open interval {(ai,bi)}i=1n on [a,b], when ∑i=1n(bi-ai)<δ, we have
(8)∑i=1n|f(bi)-f(ai)|<ε.
Obviously, if f(x) is Lipschitz continuous on [a,b], f(x) is absolutely continuous on [a,b].One of the main tool, used in the present paper, in order to prove the lower semicontinuity of the functionalF(u,Ω) in (1), is an approximation result for convex functions due to De Giorgi [10].Lemma 6 (see [10]).
LetU⊆Rd be an open set and f:U×Rn→[0,+∞) a continuous function with compact support on U, such that, for every t∈U,f(t,·) is convex on Rn. Then there exists a sequence {ηq}q=1∞⊆Cc∞(Rn),ηq≥0,∫Rn‍ηqdρ=1, and supp(ηq)⊆B(0,1), such that, if we let
(9)aq(t)=∫Rn‍f(t,ρ){(n+1)ηq(ρ)+〈∇ηq(ρ),ρ〉}dρ,bq(t)=-∫Rn‍f(t,ρ)∇ηq(ρ)dρ,
one has
(10)fj(t,ξ)=max1≤q≤j{0,aq(t)+〈bq(t),ξ〉},j∈N,
satisfying the following results:(i)
for everyj∈N,fj:U×Rn→[0,+∞) is a continuous function with compact support on U such that, for allt∈U,fj(t,·) is convex on Rn. Moreover, for all (t,ξ)∈U×Rn,fj(t,ξ)≤fj+1(t,ξ), and
(11)f(t,ξ)=supj∈Nfj(t,ξ),(ii)
for everyj∈N, there exists a constant Mj>0, such that, for all (t,ξ)∈U×Rn,
(12)|fj(t,ξ)|≤Mj(1+|ξ|),
and, for all t∈U, and ξ1,ξ2∈Rn;
(13)|fj(t,ξ1)-fj(t,ξ2)|≤Mj|ξ1-ξ2|.


## 3. Proof of Theorem4

We will divide four steps to complete the proof of Theorem4.Step 1.
Let{βi(x,s)}i∈N be a sequence of smooth functions satisfying (1)
there exists a compact setΩ′×H⊂⊂Ω×R, such that βi(x,s)=0, for all (x,s)∈(Ω∖Ω′)×(R∖H);(2)
for everyi∈N,βi(x,s)≤βi+1(x,s), for all (x,s)∈Ω′×H;(3)
limi→+∞βi(x,s)=1, for all (x,s)∈Ω′×H.Let(14)fi(x,s,ξ)=βi(x,s)f(x,s,ξ),i=1,2,….It is clear that, for eachi∈N,fi satisfies all the hypotheses in Theorem 4 and also vanishes if (x,s) is outside Ω′×H; thus
(15)limi→+∞fi(x,s,ξ)=f(x,s,ξ),∀(x,s,ξ)∈Ω′×H×R,fi(x,s,ξ)≤fi+1(x,s,ξ)≤f(x,s,ξ),∀i∈N,∀(x,s,ξ)∈Ω′×H×R.
By Levi’s Lemma, we have
(16)limi→+∞∫Ω′‍fi(x,s,ξ)dx=∫Ω′‍f(x,s,ξ)dx.
Thus, without loss of generality, we can assume that there exists an open set Ω′×H⊂⊂Ω×R, such that
(17)f(x,s,ξ)=0,∀(x,s,ξ)∈(Ω∖Ω′)×(R∖H)×R.Letum,u∈Wloc1,1(Ω) such that um→u in Lloc1(Ω). We will prove that
(18)liminfm→+∞F(um,Ω)≥F(u,Ω).
Without loss of generality, we can assume that
(19)liminfm→+∞F(um,Ω)=limm→+∞F(um,Ω)<+∞.
By (17), we have F(um,Ω)=F(um,Ω′) and F(u,Ω)=F(u,Ω′); thus we will only prove the following inequality:
(20)limm→+∞F(um,Ω′)≥F(u,Ω′).Step 2.
Letηε∈Cc∞(R) be a mollifier, and, for ϵ>0, define
(21)vε(x)=ηε*u(x)=∫Ω‍ηε(x-y)u(y)dy,x∈[Ωε],
where [Ωε]≜{x∈Ω:dist(x,∂Ω)>ε}. We have
(22)[uε(x)]′=[∫Ω‍ηε(x-y)u(y)dy]x=∫Ω‍[ηε(x-y)]xu(y)dy=∫B(x,ε)‍ηε(x-y)[u(y)]ydy=[u′]ε(x),x∈Ωε.
In the following, we denote the derivative of uε by uε′. When u∈Wloc1,1(Ω), we know u′∈Lloc1(Ω). By the properties of convolution, we know uε′∈C0∞(Ω) and
(23)uε′⟶u′inLloc1(Ω)asε⟶0+,
That is, forallδ>0,∃ϵ>0, such that
(24)∫Ω′‍|uε′-u′|dx<δ.Now we estimate the difference for the integrand values on different points:(25)f(x,um,um′)-f(x,u,u′)=f(x,um,um′)-f(x,um,uε′)+f(x,um,uε′)-f(x,u,uε′)+f(x,u,uε′)-f(x,u,u′).
By the convexity of f(x,s,ξ) with respect to ξ, we have
(26)f(x,um,um′)-f(x,um,uε′)≥fξ(x,um,uε′)·(um′-uε′).=fξ(x,um,uε′)·um′-fξ(x,u,uε′)·u′+fξ(x,u,uε′)·(u′-uε′)+[fξ(x,u,uε′)-fξ(x,um,uε′)]·uε′.
By (25) and (26), we have
(27)∫Ω′‍[f(x,um,um′)-f(x,u,u′)]dx≥∫Ω′‍[fξ(x,um,uε′)·um′-fξ(x,u,uξ′)·u′]dx+∫Ω′‍[fξ(x,u,uε′)·(u′-uε′)]dx+∫Ω′‍[fξ(x,u,uε′)-fξ(x,um,uε′)]·uε′dx+∫Ω′‍[f(x,um,uε′)-f(x,u,uε′)]dx+∫Ω′‍[f(x,u,uε′)-f(x,u,u′)]dx.Step 3.
Now, we estimate the right side of inequality (27).
By (6) and (24), we have
(28)∫Ω′‍[fξ(x,u,uε′)·(u′-uε′)]dx≥-L1∫Ω′‍|u′-uε′|dx≥-L1δ.
Thus
(29)limε→0∫Ω′‍[fξ(x,u,uε′)·(u′-uε′)]dx≥0.
Since f(x,s,ξ) and fξ(x,s,ξ) are continuous functions, they are bounded functions on compact subset. By Lebesgue Dominated Convergence Theorem, we obtain
(30)limm→+∞∫Ω′‍[fξ(x,u,uε′)-fξ(x,um,uε′)]·uε′dx=0,limm→+∞∫Ω′‍[f(x,um,uε′)-f(x,u,uε′)]dx=0.
Now, we will prove
(31)limε→0∫Ω′‍[f(x,u,uε′)-f(x,u,u′)]dx≥0.
By Lemma 6, there exists a sequence of nonnegative continuous functions fj(x,s,ξ)(j∈N), such that fj(x,s,ξ) is convex on ξ, and, for all (x,s,ξ)∈Ω′×H×R,
(32)fj(x,s,ξ)≤fj+1(x,s,ξ),f(x,s,ξ)=supj∈Nfj(x,s,ξ),|fj(x,s,ξ1)-fj(x,s,ξ2)|≤Mj|ξ1-ξ2|.
By Levi’s Lemma, we obtain
(33)limj→+∞∫Ω′‍fj(x,u,uε′)dx=∫Ω′‍f(x,u,uε′)dx,limj→+∞∫Ω′‍fj(x,u,u′)dx=∫Ω′‍f(x,u,u′)dx.
In order to prove (31), we only need to prove
(34)limε→0∫Ω′‍[fj(x,u,uε′)-fj(x,u,u′)]dx≥0,∀j∈N.
By (33), we have
(35)∫Ω′‍[fj(x,u,uε′)-fj(x,u,u′)]dx≥-Mj∫Ω′‍|uε′-u′|dx≥-Mjδ.
Thus (31) holds.Step 4.
Now, we need to prove(36)limm→+∞∫Ω′‍[fξ(x,um,uε′)·um′-fξ(x,u,uε′)·u′]dx=0.
Let
(37)g(x,s)≜fξ(x,s,uε′),x∈Ω′,(38)Gm(x)≜∫u(x)um(x)‍g(x,s)ds,x∈Ω′.
By triangle inequality and (7), we have
(39)|fξ(yi,s,uε′(yi))-fξ(xi,s,uε′(xi))|≤|fξ(yi,s,uε′(yi))-fξ(xi,s,uε′(yi))|+|fξ(xi,s,uε′(yi))-fξ(xi,s,uε′(xi))|≤|fξ(yi,s,uε′(yi))-fξ(xi,s,uε′(yi))|+L1|uε′(yi)-uε′(xi)|.
By (39), condition (H2) and uε′∈C0∞(Ω), we know that g(x,s) is a locally absolute continuous function about x. So g(x,s) is almost everywhere differentiable; that is, ∂g/∂x exists almost everywhere. Taking derivatives in both sides of (38), we obtain
(40)Gm′(x)=g(x,um)·um′-g(x,u)·u′+∫u(x)um(x)‍∂g∂xds,a.e.x∈Ω′.
Because Gm(x) vanishes outside Ω′, we obtain
(41)∫Ω′‍Gm′(x)dx=0.
By (40), we have
(42)|∫Ω′‍[fξ(x,um,uε′)·um′-fξ(x,u,uε′)·u′]dx|=|∫Ω′‍[g(x,um)·um′-g(x,u)·u′]dx|=|-∫Ω′‍∫u(x)um(x)‍∂g∂xdsdx|≤∫Dm‍|∂g∂x|dxds,
where
(43)Dm={(x,s)∈Ω′×H∣min{um(x),u(x)}≤s(x)≤max{um(x),u(x)}}.
We note
(44)|Dm|=|∫Ω′‍∫uum‍dsdx|≤∫Ω′‍|um-u|dx⟶0(m⟶+∞).
By Fubini’s Theorem, we have
(45)∫Ω×R‍|∂g∂x|dxds=∫H‍ds∫Ω′‍|∂g∂x|dx.
Since g(x,s) is absolutely continuous about x, ∂g/∂x is integrable and absolutely integrable with respect to x; that is,
(46)∫Ω′‍|∂g∂x|dx<+∞.
By (17) and (46), we obtain
(47)∫Ω×R‍|∂g∂x|dxds<+∞.
Because of the absolute continuity of integral, we have
(48)limm→+∞∫Dm‍|∂g∂x|dxds=0.
By (42), we obtain
(49)limm→+∞|∫Ω′‍[fξ(x,um,uε′)·um′-fξ(x,u,uε′)·u′]dx∫Ω′‍|=0.
Thus we just proved (36). By (29)–(31) and (36), we have
(50)limm→+∞∫Ω′‍[f(x,um,um′)-f(x,u,u′)]dx≥0.
Thus we deduce that the functional F(u,Ω) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1(Ω). We complete the proof.


## 4. Proof of Theorem5

In order to prove Theorem5, we will verify all the conditions in Theorem 4 under the assumptions in Theorem 5. Now we will divide three steps to complete the proof of Theorem 5.Step 1.
Similar to the first step of the proof in Theorem4, without loss of generality, we assume that the integrand f(x,s,ξ) vanishes outside a compact subset of Ω×R. Thus we assume that there exists an open set Ω′×H⊂⊂Ω×R, such that
(51)f(x,s,ξ)≡0,∀(x,s,ξ)∈(Ω∖Ω′)×(R∖H)×R.
Letum,u∈Wloc1,1(Ω), such that um→u in Lloc1(Ω); we need to prove
(52)limm→+∞F(um,Ω′)≥F(u,Ω′).
By Lemma 6, there exists a function sequence {fj(x,s,ξ)}j∈N, such that, for all j∈N, fj is a continuous function on Ω′×H⊂⊂Ω×R, for all (x,s)∈Ω′×H,  fj(x,s,·) is convex on R, and, forall(x,s,ξ)∈Ω′×H×R,
(53)fj(x,s,ξ)≤fj+1(x,s,ξ),(54)f(x,s,ξ)=supj∈Nfj(x,s,ξ),(55)|fj(x,s,ξ1)-fj(x,s,ξ2)|≤Mj|ξ1-ξ2|,(x,s)∈Ω′×H,ξ1,ξ2∈R.
Let ηε∈Cc∞(R)(0<ε≪1) be a mollifier, and define the fj,ε=fj*ηε; that is,
(56)fj,ε(x,s,ξ)=∫R‍fj(x,s,ξ-z)ηε(z)dz.
By (55), we have
(57)|fj,ε(x,s,ξ)-fj(x,s,ξ)|≤∫R‍|fj(x,s,ξ-z)-fj(x,s,ξ)|ηε(z)dz≤∫suppηε‍Mj|z|·ηε(z)dz≤Mj·ε.
Choose ε=εj=1/jMj→0. By (57), we have
(58)|fj,εj(x,s,ξ)-fj(x,s,ξ)|≤Mjεj=1j.
So
(59)fj(x,s,ξ)-2j≤fj,εj(x,s,ξ)-1j≤fj(x,s,ξ)≤f(x,s,ξ).
By (53), (54), and Levi’s Lemma, we have
(60)limi→+∞∫Ω′‍fj(x,u(x),u′(x))dx=∫Ω′‍f(x,u(x),u′(x))dx.
Let
(61)Fj(u,Ω′)=∫Ω′‍[fj,εj(x,u(x),u′(x))-1j]dx.
By (59)–(61), we have
(62)limj→+∞Fj(u,Ω′)=F(u,Ω′)=∫Ω′‍f(x,u(x),u′(x))dx.
Obviously,
(63)Fj(u,Ω′)≤F(u,Ω′),∀j∈N.
Thus
(64)supj∈NFj(u,Ω′)=F(u,Ω′).
Therefore F(u,Ω′), being the supremum of the family of functionals {Fj(u,Ω′)}j∈N, will be lower semicontinuous if every {Fj(u,Ω′)} is lower semicontinuous.Step 2.
In order to prove that, for allj∈N,Fj(u,Ω′) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1(Ω), we will prove that, for all j∈N, the integrand fj,εj(x,u(x),u′(x)) satisfies all conditions of Theorem 4. Obviously, forallj∈N, fj,εj satisfies condition (H1).
For all(x,s)∈Ω′×H and ξ1,ξ2∈R, by (55), we have
(65)|fj,εj(x,s,ξ1)-fj,εj(x,s,ξ2)|≤∫R‍|fj(x,s,ξ1-z)-fj(x,s,ξ2-z)|·ηεj(z)dz≤∫suppηε‍Mj|ξ1-ξ2|ηεj(z)dz≤Mj|ξ1-ξ2|.
Thus
(66)|∂fj,εj∂ξ|≤Mj.
So fj,εj satisfies (6) in condition (H3) of Theorem 4.Now, we will prove thatfj,εj satisfies (7) in condition (H3) of Theorem 4. By supp(ηεj)⊆B(0,εj), we have
(67)∂fj,εj∂ξ(x,s,ξ)=∫R‍∂fj(x,s,ξ-z)∂ξ·ηεj(z)dz=-∫R‍∂fj(x,s,ξ-z)∂z·ηεj(z)dz=∫R‍fj(x,s,ξ-z)∂ηεj(z)∂zdz.
By (55) and (67), we have
(68)|∂fj,εj∂ξ(x,s,ξ1)-∂fj,εj∂ξ(x,s,ξ2)|≤∫R‍|fj(x,s,ξ1-z)-fj(x,s,ξ2-z)|·|∂ηεj(z)∂z|dz≤Mj|ξ1-ξ2|·∫R‍|∂ηεj(z)∂z|dz=LjMj|ξ1-ξ2|,
where
(69)Lj=∫R‍|∂ηεj(z)∂z|dz
is a constant depending on εj. Thus fj,εj satisfies (7). So we proved that fj,εj satisfies condition (H3).Step 3.
Next we will prove thatfj,εj satisfies condition (H2).By condition (H4), for every compact subsetΩ′×H×K, f(x,s,ξ) is absolutely continuous about x, that is, for all ε0>0,∃δ>0 such that for any finite disjoint open interval {(xi,yj)}i=1n in Ω′, when ∑i=1n(yi-xi)<δ, we have
(70)∑i=1n‍|f(yi,s,ξ)-f(xi,s,ξ)|<ε0.
By Step 1, {fj(x,s,ξ)}j∈N satisfies (53)-(55) and the following property:
(71)fj(x,s,ξ)=max1≤q≤j{0,aq(x,s)+bq(x,s)ξ},j∈N,
where
(72)aq(x,s)=∫R‍f(x,s,ρ)[2ηq(ρ)+ρ∂ηq(ρ)∂ρ]dρ,bq(x,s)=-∫R‍f(x,s,ρ)∂ηq(ρ)∂ρdρ,
And, for all (x,s,ξ)∈Ω′×H×R,ηq∈Cc∞(R)(q∈N) are mollifiers satisfying ηq≥0,∫R‍ηq(ρ)dρ=1, and supp(ηq)⊆B(0,1), for all j∈N. By (71), without of loss generality, we assume that there exists l∈{1,…,j}, such that
(73)fj(x,s,ξ)=al(x,s)+bl(x,s)·ξ,
where al,bl are given by (72). By (70), we obtain
(74)∑i=1n‍|al(yi,s)-al(xi,s)|≤∫R‍∑i=1n‍|f(yi,s,ρ)-f(xi,s,ρ)|·[2ηl(ρ)+|ρ∂ηl(ρ)∂ρ|]dρ≤ε0∫B(0,1)‍[2ηl(ρ)+|ρ∂ηl(ρ)∂ρ|]dρ≤(2+Al)·ε0,
where
(75)Al=∫B(0,1)‍|∂ηl(ρ)∂ρ|dρ
is a constant. Similar to the above proof, we have
(76)∑i=1n‍|bl(yi,s)-bl(xi,s)|≤∫R‍∑i=1n‍|f(yi,s,ρ)-f(xi,s,ρ)|·|∂ηl(ρ)∂ρ|dρ≤ε0∫B(0,1)‍|∂ηl(ρ)∂ρ|dρ≤Al·ε0.
Thus
(77)∑i=1n‍|fj(yi,s,ξ)-fj(xi,s,ξ)|≤∑i=1n‍|al(yi,s)-al(xi,s)|+∑i=1n‍|bl(yi,s)-bl(xi,s)|·|ξ|≤(2+Al)ε0+Alε0K1=(2+Al+AlK1)ε0≜σ.
Since ξ belongs to a compact set, then K1=supξ{|ξ|}<+∞. Choose ε0 sufficient small, so that σ is small enough. Thus fj(x,s,ξ) is absolutely continuous about x for all (x,s,ξ)∈A, which is a compact subset of Ω×R×R. By (56) and (77), we have
(78)∑i=1n‍|fj,εj(yi,s,ξ)-fj,εj(xi,s,ξ)|≤∫R‍∑i=1n‍|fj(yi,s,ξ-z)-fj(xi,s,ξ-z)|·ηεj(z)dz≤σ·∫B(0,εj)‍ηεj(z)dz=σ.
By (67) and (78), we obtain
(79)∑i=1n‍|∂fj,εj∂ξ(yi,s,ξ)-∂fj,εj∂ξ(xi,s,ξ)|≤∫R‍∑i=1n‍|fj(yi,s,ξ-z)-fj(xi,s,ξ-z)|·|∂ηεj(z)∂z|dz≤σ∫R‍|∂ηεj(z)∂z|dz=Ljσ,
where Lj are constants depending on εj and given by (69) (forallj∈N). By (79), for every compact subset on Ω×R×R,∂fj,εj/∂ξ is absolutely continuous about x. Thus fj,εj satisfies condition (H2).Now, we have provedfj,εj satisfies all conditions in Theorem 4, so Fj(u,Ω′) is lower semicontinuous in Wloc1,1(Ω) with respect to the strong convergence in Lloc1(Ω). Thus F(u,Ω) has the same lower semicontinuity. This completes the proof of Theorem 5.



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*Source: 368610-2013-06-20.xml*