Local Lipschitz continuity of solutions to a problem - Pierre Bousquet

16 Pages·2007·0.2 MB·English

Local Lipschitz continuity of solutions to a problem
in the calculus of variations
Pierre Bousquet∗ and Francis Clarke†
December 2006
Dedicated to Arrigo Cellina
on the occasion of his 65th birthday
Abstract
R
This article studies the problem of minimizing F(Du)+G(x,u)
Ω
over the functions u ∈ W1,1(Ω) that assume given boundary values φ
on∂Ω. ThefunctionF andthedomainΩareassumedconvex. Incon-
sideringthesameproblemwithG=0,andinthespiritoftheclassical
Hilbert-Haar theory, Clarke has introduced a new type of hypothesis
ontheboundaryfunctionφ: thelower (orupper)bounded slope condi-
tion. Thiscondition,whichislessrestrictivethantheclassicalbounded
slopeconditionofHartman,NirembergandStampacchia,issatisfiedif
φ is the restriction to ∂Ω of a convex (or concave) function. We show
that for a class of problems in which G(x,u) is locally Lipschitz (but
notnecessarilyconvex)inu,thelowerboundedslopeconditionimplies
the local Lipschitz regularity of solutions.
1 Introduction.
We study the regularity of solutions to the following problem (P) in
the multiple integral calculus of variations:
Z
min {F(Du(x))+G(x,u(x))}dxsubject tou∈W1,1(Ω), tru=φ,
u
Ω
where Ω is a domain in Rn, u is scalar-valued, and tru signifies the
trace of u on Γ:=∂Ω.
∗Universit´e Claude Bernard Lyon 1, <[email protected]>.
†Membre de l’Institut universitaire de France et professeur `a l’Universit´e Claude
Bernard Lyon 1 <[email protected]>.
1
1 INTRODUCTION. 2
The aim is to deduce local Lipschitz regularity from properties of
the boundary function φ. This is in the general spirit of the well-
known Hilbert-Haar theory (see for example [5][10]), which requires
that φ satisfy the bounded slope condition (BSC). The BSC of rank K
is the assumption that, given any point γ ∈ Γ, there exist two affine
functions
y 7→hζ−,y−γi+φ(γ), y 7→hζ+,y−γi+φ(γ)
γ γ
agreeing with φ at γ, whose slopes satisfy |ζ−| ≤ K, |ζ+| ≤ K, and
γ γ
such that
hζ−,γ0−γi+φ(γ)≤φ(γ0)≤hζ+,γ0−γi+φ(γ) ∀γ0 ∈Γ.
γ γ
The classical Hilbert-Haar theorem asserts that if F is convex, G =
0, and φ satisfies the BSC, then there exists a (globally) Lipschitz
minimizer for (P). The first proof of this statement is due to Miranda
[9], although there are several special cases that are antecedents to
this. The case in which G is different from 0 has been treated by
Stampacchia [11] (and implicitly in [7]) under stronger smoothness
assumptions on the data than used here. As regards other and more
recent uses of the BSC, see notably Cellina [2] and other references
cited therein.
The BSC is a restrictive requirement on flat parts of Γ, since it
forces φ to be affine. Moreover, if Ω is smooth, then it forces φ to
be smooth as well (see Hartman [6] for precise statements). Recently,
Clarke [3] has introduced a new hypothesis on φ, the lower bounded
slopecondition (LBSC)ofrankK :givenanypointγ ontheboundary,
there exists an affine function
y 7→hζ ,y−γi+φ(γ)
γ
with |ζ |≤K such that
γ
hζ ,γ0−γi+φ(γ)≤φ(γ0) ∀γ0 ∈Γ.
γ
This requirement, which can be viewed as a one-sided BSC, enlarges
considerablytheclassofboundaryfunctionswhichitallows(compared
totheBSC).ThepropertyhasbeenstudiedbyBousquetin[1], where
it is shown that φ satisfies the LBSC if and only if it is the restriction
to Γ of a convex function. When Ω is uniformly convex, φ satisfies the
LBSC if and only if it is the restriction to Γ of a semiconvex function.
It turns out that the LBSC has significant implications for the
regularity of the solution u, although it implies less than the full, two-
sided BSC. In fact, it is shown in [3] that in the case where G = 0,
theone-sidedBSCgivesthecrucialregularitypropertythatoneseeks:
u is locally Lipschitz in Ω. This allows one to assert that u is a weak
2 THE MAIN RESULT. 3
solutionoftheEulerequation,intheabsenceoftheusualuppergrowth
conditions on F. Furthermore, the local Lipschitz property allows one
to invoke De Giorgi’s regularity theory (when the data are sufficiently
smooth) to obtain the continuous differentiability of the solution.
The goal of this article is to prove local Lipschitz regularity of
the solution for a class of problems with G different from 0, under
weak regularity hypotheses on the data of the problem, and when the
LBSC is satisfied (rather than the BSC). The next section describes
thehypothesesandgivesaself-containedproofofthemaintheoremof
the article. It is most closely related to the work of Stampacchia, but
the method of proof differs in several important respects. A variant
of the main theorem is developed in Section 3, and the final section
discusses the issue of the continuity of the solution at the boundary.
2 The main result.
We now specify the hypotheses on the data of the problem (P). The
first one, in particular, justifies the use of trace.
(HΩ) Ω is an open bounded convex set.
We require that F be uniformly elliptic, and that G be locally Lip-
schitz in u. More precisely:
(HF) For some µ>0, F satisfies, for all θ ∈(0,1) and p,q ∈Rn:
θF(p)+(1−θ)F(q)≥F(θp+(1−θ)q)+(µ/2)θ(1−θ)|p−q|2.
We remark that when F is of class C2, (HF) holds if and only if, for
every v ∈Rn, we have
(cid:10)z,∇2F(v)z(cid:11)≥µ|z|2 ∀z ∈Rn.
R
Under(HF),itiseasytoseethat F(Dw)dxiswell-defined(possibly
Ω
as +∞) for any w ∈W1,1(Ω).
(HG) G(x,u)ismeasurableinxanddifferentiableinu,andforevery
bounded interval U in R there is a constant L such that for almost all
x∈Ω,
|G(x,u)−G(x,u0)|≤L|u−u0| ∀u,u0 ∈U.
We also postulate as part of (HG) that for some bounded function b,
R
the integral G(x,b(x))dx is well-defined and finite. It follows that
Ω
the same is true for all bounded measurable functions w.
2 THE MAIN RESULT. 4
In the presence of (HΩ),(HF), and (HG), it follows that
Z
I(w):= {F(Dw(x))+G(x,w(x))}dx
Ω
is well-defined for all w ∈ W1,1(Ω) for which w is bounded. We say
that u solves (P) relative to L∞(Ω) if u is itself bounded, and if we
have I(u)≤I(w) for all bounded w that are admissible for (P).
The theorem to be proved is the following.
Theorem 2.1
Under the hypotheses (HΩ),(HF), and (HG), and when φ satisfies
the Lower Bounded Slope Condition, any solution u of (P) relative to
L∞(Ω) is locally Lipschitz in Ω.
In the context of the theorem, even when G = 0 and F(v) =
|v|2, a bounded solution u of (P) may fail to be globally Lipschitz;
an example of this type is given in [1][3]. Let us also point out that
thetheoremhasanalternateversioninwhichtheLBSCisreplacedby
the upper BSC; the conclusion is the same. Finally, we remark that
Stampacchia [11] has described structural assumptions on G which
guarantee a priori the existence and boundedness of solutions of (P);
these will be described in the next section.
2.1 The lower barrier condition
The proof of the main result uses in part the well-known barrier tech-
nique. Our one-sided version of this is the following.
Theorem 2.2
Under hypotheses (HΩ),(HF), and (HG), let u be a bounded solu-
tion of problem (P) as described above, where φ satisfies the Lower
Bounded Slope Condition of rank K. Then there exists K¯ > 0 with
the following property: for any γ ∈ Γ there exists a function w which
is Lipschitz of rank K¯, which agrees with φ at γ, and which satisfies
w ≤u a.e. in Ω.
Proof We may suppose that φ is a globally defined convex function
of Lipschitz rank K. Thus there is an element ζ with |ζ| ≤ K in the
subdifferential of φ at γ:
φ(x)−φ(γ)≥hζ,x−γi ∀x∈Rn.
By (HG) there is a Lipschitz constant L valid for G(x,·) over the
interval
(cid:2) (cid:3)
−kuk ,kφk +KdiamΩ ,
L∞(Ω) L∞(Γ)
2 THE MAIN RESULT. 5
for x ∈ Ω a.e. Fix any T > (L+1)exp(diamΩ)/µ, where µ is given
by (HF).
ThefollowingconstructionisarefinementofthatproposedbyHart-
mann and Stampacchia [7] (Lemma 10.1). Let ν be a unit outward
normal vector to Ω at γ, and define
w(x):=φ(γ)+hζ,x−γi−T{1−exp(hx−γ,νi)}.
We proceed to prove that w has the required properties. Clearly w
agrees with φ at γ, and is Lipschitz of rank
K¯ :=K+T exp(diamΩ).
We need only show that the set
S :={x∈Ω:w(x)>u(x)}
has measure 0.
The function M(x):=max[u(x),w(x)] belongs to W1,1(Ω) (see for
example [4] or [8]), and we have:
DM(x)=Dw(x),x∈S a.e., DM(x)=Du(x),x∈Ω\S a.e.
It follows from the subgradient inequality for ζ that M ∈φ+W1,1(Ω)
0
(in deriving this, we also use the fact that hx−γ,νi ≤ 0 for x ∈ Ω).
By the optimality of u (relative to M) we deduce
Z Z
{F(Du(x))+G(x,u(x))}dx≤ {F(Dw(x))+G(x,w(x))}dx.
S S
The Lipschitz condition satisfied by G now leads to
Z Z
{F(Du(x))−F(Dw(x))}dx≤L {w(x)−u(x)}dx (1)
S S
Inderivingthenextestimate(whichconcludestheproof), letusmake
the temporary assumption that F is smooth (C2 or better). Then, by
straightforwardcalculation, thefunctionψ(x):=∇F(Dw(x))satisfies
divψ(x)=T exp(hx−γ,νi)(cid:10)ν,∇2F(Dw(x))ν(cid:11)≥L+1, (2)
in light of (HF), and because of how T was chosen. We proceed to
deduce from (1) the following:
Z Z
L {w(x)−u(x)}dx≥ {F(Du(x))−F(Dw(x))}dx
S S
Z
≥ hψ(x),Du(x)−Dw(x)i dx
S
2 THE MAIN RESULT. 6
(by the subdifferential inequality)
Z
= hψ(x),Dmin[u,w](x)−Dw(x)i dx
Ω
Z
= (divψ(x))(w(x)−min[u,w](x))dx
Ω
(integration by parts, noting that min[u,w]=w on Γ)
Z
≥(L+1) {w(x)−min[u,w](x)}dx
Ω
(in view of (2))
Z
≥(L+1) {w(x)−u(x)}dx.
S
This shows that S is of measure 0, since w−u>0 in S.
In the general case in which F is not smooth, we consider a non-
decreasing sequence {Fk}k∈N of functions in C∞(Rn) converging to F
uniformly on bounded sets, and such that the ellipticity condition in
(HF) holds for F when p,q are restricted to a ball B(0,K +1) con-
k
tainingallthevaluesofDw. Suchasequenceexistsbyamollification-
truncation argument; see Morrey [10], Lemma 4.2.1. Then, arguing as
above, we derive, for any k ≥1,
Z Z
{F (Du(x))−F (Dw(x))}dx≥(L+1) {w(x)−u(x)}dx.
k k
S S
The result now follows from the Monotone Convergence Theorem. 2
2.2 Proof of Theorem 2.1.
Let λ and q be parameters satisfying
λ∈[1/2,1), q >q¯:=K¯ diamΩ+||φ|| ,
L∞(Γ)
and fix any point z ∈Γ. We denote
Ω :=λ(Ω−z)+z.
λ
Note that Ω is a subset of Ω, since the latter is convex. We proceed
λ
to define the following function on Ω :
λ
u (x):=λu((x−z)/λ+z)−q(1−λ).
λ
Then u belongs to W1,1(Ω )+φ , where
λ 0 λ λ
φ (y):=λφ((y−z)/λ+z)−q(1−λ).
λ
For every x∈Rn, we will denote (x−z)/λ+z by x .
λ
We are now going to compare u and u on Γ := ∂Ω ; this com-
λ λ λ
parison via dilation was introduced in [3].
2 THE MAIN RESULT. 7
Lemma 1. We have u ≤u on Γ .
λ λ
The meaning of this inequality is that (u −u)+ :=max(0,u −u)
λ λ
belongs to W1,1(Ω ), where here u signifies of course the restriction of
0 λ
utoΩ . ToprovetheLemma,recallfirstthatintheprecedingsection
λ
we proved the existence, for any γ ∈ Γ, of a K¯-Lipschitz function w
γ
such that w (γ) = φ(γ) and w ≤ u a.e. in Ω (which implies w ≤ φ
γ γ γ
on Γ).
Introduce l(y) := sup w (y). Then l is a K¯-Lipschitz function
γ∈Γ γ
which coincides with φ on Γ and which has l ≤ u a.e. on Ω. Thus
u−l ∈W1,1(Ω). There exists therefore a sequence v ∈Lip (Ω) (the
0 m 0
class of Lipschitz functions vanishing at the boundary) converging to
u−linW1,1(Ω)andalmosteverywhereinΩ.Wecansupposemoreover
v ≥0, by replacing v by v+ :=max(v ,0). We have used here the
m m m m
fact that if a sequence of functions k converges almost everywhere
m
and in W1,1(Ω) to k, then k+ converges to k+ in W1,1(Ω).
m
We define the functions
u (x):=v (x)+l(x) , u (x):=λu ((x−z)/λ+z)−q(1−λ).
m m m,λ m
(Notethatu isdefinedonΩ,andu onΩ .) Theseregularizations
m m,λ λ
of u and u will allow us to complete the proof of the Lemma.
λ
Wehaveu ∈C0(Ω¯),l≤u onΩandu =φ=l onΓ.Weclaim
m m m
that u (γ)≤u (γ) for every m≥0,γ ∈Γ . Suppose for a moment
m,λ m λ
this claim were true. Then we could assert that
(u −u )+ ∈W1,1(Ω ).
m,λ m 0 λ
Now, u tends to u in W1,1(Ω ) and almost everywhere, as does
m,λ λ λ
u to u. It would follow therefore that (u −u)+ ∈ W1,1(Ω ), which
m λ 0 λ
is what we wish to prove.
So it suffices to prove the claim. Fix some γ ∈Γ . Then,
λ
u (γ)−u (γ)=λu (γ )−u (γ)−q(1−λ)
m,λ m m λ m
≤λφ(γ )−l(γ)−q(1−λ)
λ
=λl(γ )−l(γ)−q(1−λ)
λ
≤(l(γ )−l(γ))+(1−λ)(||l|| −q)
λ L∞(Γ)
≤K¯|γ−γ |+(1−λ)(||l|| −q)
λ L∞(Γ)
≤(1−λ)(K¯ diamΩ+||l|| −q)
L∞(Γ)
≤0,
since γ ∈Γ, ||l|| =kφk , and because q has been chosen to
λ L∞(Γ) L∞(Γ)
be greater than q¯. This proves the claim and completes the proof of
Lemma 1.
2 THE MAIN RESULT. 8
The next step of the proof is to show that the set
A:={y ∈Ω :u (y)>u(y)}
λ λ
hasmeasurezero. Letw(x):=min(u,u ),whichbelongstoW1,1(Ω )+
λ 0 λ
φ in light of Lemma 1, and define
λ
1
w˜λ(x):= w(λ(x−z)+z)+q(λ−1−1),
λ
anelementofW1,1(Ω)+φ. Fixanyθ ∈(0,1). Thenv :=θw˜λ+(1−θ)u
0
lies in W1,1(Ω)+φ, so that I(u)≤I(v), which yields after an evident
0
change of variables
Z n y−z u +q(1−λ) o
F(Du )+G( +z, λ ) dy ≤
λ λ λ
Ωλ
Z n y−z w+q(1−λ)
F(θDw+(1−θ)Du )+G( +z,θ +
λ λ λ
Ωλ
u +q(1−λ) o
(1−θ) λ ) dy.
λ
R
We also note that the right side is finite, since F(Du)dx is finite,
Ω
and in light of the convexity of F. This implies
Z n y−z u +q(1−λ) o
F(Du )+G( +z, λ ) dy ≤
λ λ λ
A
Z n y−z w+q(1−λ)
F(θDw+(1−θ)Du )+G( +z,θ +
λ λ λ
A
u +q(1−λ) o
(1−θ) λ ) dy,
λ
whence (since w =u on A)
Z
{F(Du )−F(θDu+(1−θ)Du )}dy ≤
λ λ
A
Z n y−z u+q(1−λ) u +q(1−λ)
G( +z,θ +(1−θ) λ )
λ λ λ
A
y−z u +q(1−λ) o
−G( +z, λ ) dy. (3)
λ λ
Now let W(x) := max(u(x),u (x)) for x ∈ Ω , and W(x) := u(x)
λ λ
for x ∈ Ω\Ω . Then W ∈ W1,1(Ω)+φ since u ≤ u on Γ . With
λ 0 λ λ
v :=θW +(1−θ)u, we have I(u)≤I(v), which yields
Z
{(F(Du)+G(y,u))}dy ≤
A
Z
{(F(θDu +(1−θ)Du)+G(y,θu +(1−θ)u))}dy,
λ λ
A
2 THE MAIN RESULT. 9
so that
Z
{(F(Du)−F(θDu +(1−θ)Du))}dy ≤
λ
A
Z
{(G(y,θu +(1−θ)u)−G(y,u))}dy. (4)
λ
A
Summing (3) and (4), we get
Z n
(1−θ)F(Du )+θF(Du)−F(θDu+(1−θ)Du )
λ λ
A
o
+θF(Du )+(1−θ)F(Du)−F(θDu +(1−θ)Du) dy
λ λ
Z n y−z u+q(1−λ) u +q(1−λ)
≤ G( +z,θ +(1−θ) λ )
λ λ λ
A
y−z u +q(1−λ)
−G( +z, λ )
λ λ
o
+G(y,θu +(1−θ)u)−G(y,u) dy. (5)
λ
Thanks to (HF) we see that the left side of the last inequality is no
less than
Z
µθ(1−θ) |Du −Du|2dy.
λ
A
Substituting into (5), dividing by θ, and letting θ go to 0, we find
Z Z 1 y−z
µ |Du −Du|2dy ≤ ( g( +z)−g(y))(u −u)dy,
λ λ λ λ
A A
where we have denoted by g(y) the function −G (y,u(y)), which be-
u
longs to L∞(Ω). Write for any y ∈Ω :
λ
1 y−z 1 1 y−z
g( +z)−g(y)=( − )g( +z)+h(y)
λ λ λ λn λ
where we define h(y) := 1/λng((y−z)/λ+z)−g(y) for y ∈ Ω and
λ
h(y)=0 for y ∈Rn\Ω . Then
λ
Z Z 1 1
µ |Du −Du|2dy ≤ [( − )g +h(y)](u −u)dy, (6)
λ λn λ 0 λ
A A
where g :=||g|| .
0 ∞
Lemma 2. There exists f = (f ,...,f ) ∈ L∞(Rn)n such that
1 n
for each j ∈ {1,...,n}, for almost every (y ,...,y ,y ,...,y ) ∈
1 j−1 j+1 n
Rn−1, the function
y 7→f (y ,...,y ,y ,y ,...,y )
j j 1 j−1 j j+1 n
2 THE MAIN RESULT. 10
is absolutely continuous on R with
∂f
j ∈L∞(Rn)
∂y
j
and
n
divf :=X∂fj =h a.e.inΩ (7)
∂y λ
j
j=1
and such that ||f|| ≤ C (1 − λ) for some constant C which
L∞(Ω) 0 0
depends only on g and Ω (λ being restricted to [1/2,1)).
0
Proof of Lemma 2. We extend g by setting it equal to 0 outside
Ω. There exists c=(c ,...,c )∈Rn such that
1 n
Ω⊂{(x ,...,x )∈Rn :x ≥c ∀j =1,...,n}.
1 n j j
Define f (y ,...,y ) to be
1 1 n
Z y1(cid:20)1 y0 −z (cid:21)
g( 1 1 +z ,y ,...,y )−g(y0,y ,...,y ) dy0
λ λ 1 2 n 1 2 n 1
c1
and similarly set f (y ,...,y ) equal to
j 1 n
Z yjh 1 g(y1−z1 +z ,...,yj0 −zj +z ,y ,...,y )idy0−
λj λ 1 λ j j+1 n j
cj
Z yjh 1 y −z y −z i
g( 1 1+z ,..., j−1 j−1+z ,y0,y ,...,y ) dy0
λj−1 λ 1 λ j−1 j j+1 n j
cj
This implies
∂f
j ∈L∞(Rn)
∂y
j
and that f satisfies (7). Upon making the change of variables y00 =
j
(y0 −z )/λ+z in the first integral defining f , we find readily
j j j j
1 y −z c −z
|f (y ,...,y )|≤g {| j j +z −y |+| j j +z −c |}
j 1 n 0λj−1 λ j j λ j j
1−λ
=g {|y −z |+|c −z |}
0 λj j j j j
1−λ
≤g {diamΩ+|c|+ max|z|},
0 λj z∈Ω
if the y lie in Ω. This completes the proof of the Lemma.
i