A geometric approach for convexity in some variational problem in the Gauss space

A Geometric Approach for Convexity in Some Variational Problem in the Gauss Space

M. GOLDMAN(*)

ABSTRACT- In this short note we prove the convexity of minimizers of some varia- tional problem in the Gauss space. This proof is based on a geometric version of an older argument due to Korevaar.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 49N60; 60D05; 35J70.

KEYWORDS. Convexity - obstacle problem - regularity - total variation - Gauss space.

1. Introduction

In the paper [5] we prove together with A. Chambolle and M. Novaga, the convexity of the minimizers of the following variational problem in the Wiener spaceX:

min Z

X

F(ru)‡(u g)² 2 dg …1†

under the general hypothesis thatFandgare convex functions. The idea is to approximate the infinite dimensional problem by a finite dimensional one, to show convexity of the minimizers of the finite dimensional problems and prove convergence of these minimizers towards the minimum of (1). In [5], we followed the approach of Alvarez Lasry and Lions [1] to prove convexity in finite dimension. The aim of this note is to show an alternative proof based on ideas of Korevaar [10] whenFis the total variation (which was our main motivation in [5]). More precisely, we will show that for

(*) Indirizzo dell'A.: CMAP, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau, France.

E-mail: [email protected]

g2L²_g(R^m) a convex function then the solution of

BVg(Rmin^m)\L²g(R^m)

Z

R^m

jD_guj ‡1 2

Z

R^m

ju gj²dg …2†

is convex. As a by-product of our analysis we will also get that the mini- mizers of the Ornstein-Uhlenbeck functional

Hmin¹_g(R^m)

Z

R^m

jruj² 2 dg‡1

2 Z

R^m

ju gj²dg

are convex ifgis convex.

The plan of the note is the following. In Section 2 we recall some no- tation about functions of bounded variation and in Section 3 we show the convexity of the minima of (2).

2. Notation and preliminary results

Letm2Nbe fixed and letgbe the standard Gaussian measure onR^m. Let us denote byL²_g(R^m):ˆL²(R^m;g).

We now give the definitions of Sobolev spaces and functions of bounded variation in the Gauss space. For a smooth function F:R^m!R^m, we define

divgF(x):ˆdivF Fx:

The operator divgis the adjoint of the gradient for theL²_g(R^m) duality so that for every u2 C¹_c(R^m) and every F2 C¹_c(R^m;R^m), the following in- tegration by parts holds:

Z

R^m

udivgFdgˆ Z

R^m

ruFdg:

…3†

The gradient is a closable operator and we will denote byH¹_g(R^m) the do- main of its closure in L²_g(R^m). From this, formula (3) still holds for u2H¹_g(R^m) andF2 C¹_c(R^m;R^m).

Givenu2L¹_g(R^m) we say thatu2BV_g(R^m) if Z

R^m

jDguj ˆsup Z

R^m

udivgFdg; F2 C¹_c(R^m;R^m);jF(x)j 18x2R^m 8<

:

9=

;5‡1:

We see that functions in BVg(R^m) are in BVloc(R^m) and that DguˆgDu so that most of the properties of classicalBV functions ex- tends to functions in BVg(R^m) (see [2]). In particular for every setE of finite Gaussian perimeter, the reduced boundary @Eof Eis rectifiable and every point of this reduced boundary has an outward normal n^E. Defining the sets E^(s) by

E^(s)ˆ x2R^m : lim

r!0

jE\B_r(x)j jBr(x)j ˆs

then we haveH^{m 1}((E⁽⁰⁾[E⁽¹⁾[@E)^c)ˆ0 for every set of finite Gaussian perimeter. Here we denoted by H^{m 1} the m 1 dimensional Hausdorff measure.

We finally recall some facts about pairings between measures and bounded functions (see [4] for more details).

We define the space X2 to be the space of bounded functions z with divgz2L²_g(R^m) (where the divergence is intended in the distributional sense). For every smooth open setV, the trace [zn] can be defined in such a way that the integration by part formula

Z

V

(z;Dgu)‡ Z

V

udivgzdgˆ Z

@V

[zn]ugdH^{m 1}

holds forz2X2andu2BVg\L²_g(R^m) where as usual (z;Dgu) is the mea- sure defined by

Z

R^m

W(z;D_gu) ˆ Z

R^m

uWdiv_gz dg Z

R^m

uz rWdg

for everyW2 C¹_c(R^m;R^m).

3. Convexity of the minimizer

In this section we are going to prove the following result: ifg2L²_g(R^m) be a convex function then the minimizer of

BVgmin\L²_g(R^m)

Z

R^m

jDguj ‡1 2

Z

R^m

ju gj²dg …4†

is a convex function.

As in many other papers involving the total variation, we are going to study first the regularized problem associated to the functional:

minBVg Je(u)ˆ Z

R^m



e²‡ jDguj² q

‡1 2

Z

R^m

ju gj²dg …5†

where as usual, if the Radon-Nikodym decomposition of Dgu is given by Dguˆ rudg‡D^s_guwe let

Z

R^m



e²‡ jD_guj² q

ˆ Z

R^m



e²‡ jruj² q

dg‡ jD^s_guj(R^m):

As a simple consequence of the Reshetnyak's continuity Theorem we have that J_e is lower semicontinuous for the L²_g(R^m) convergence (see [2]).

We start by studying the Dirichlet problem on balls, namely

BVg(BminR)\L²_g(R^m)

Z

BR



e²‡ jDguj² q

‡1 2

Z

BR

ju gj²dg‡ Z

@BR

ju MjgdH^{m 1}: …6†

HereBRis the open ball of radiusRcentered in the origin andMis a constant to be chosen later. The term R

@BR

ju MjgdH^{m 1}can be seen as a Dirichlet term (see [6] and [3]). In the following we will denote by F(p)ˆ 

e²‡ jpj² q

.

On bounded domains, by Theorem 6.7 in [3] we can give a character- ization of the minimizers of (6).

THEOREM 3.1 (Characterization of the minima). A function u2BVg(BR)minimizes(6)if and only if ru

e²‡ jruj²

q 2X2and

divg ru e²‡ jruj² q

0 B@

1

CA‡uˆg; ru e²‡jruj²

q D^s_guˆ jD^s_guj jD^s_guj a:e:

and

"

ru e²‡ jruj²

q n

#

2sign(M u) H^{m 1} a:e:in @B_R;

wherenis the outward normal to BRand where for a and b real numbers, by a2sign(b)we mean that if b6ˆ0, aˆsign(b)and if bˆ0, a2[0;1].

Adapting very slightly the proof of [3, Th. 5.16], we get the following comparison principle:

PROPOSITION 3.2 (Comparison). Let g1g2 and W₁W₂ then the minimizers u_iwith iˆ1;2of

BVg(BminR)\L²_g(R^m)

Z

BR

F(Dgu)‡1 2

Z

BR

ju gij²dg‡ Z

@BR

ju W_ijgdH^{m 1}

verify u₁u₂.

With this comparison property in hands, we can prove that for M large enough, the minimizer of (6) makes vertical contact angle with the boundary of B_R. In the following, we will say that a function v is a supersolution of (6) if there exists ~gg and WM such that v minimizes the functional (6) with ~g instead of g and W instead of M. By Proposition 3.2 this implies that v is larger that the minimizer of (6).

PROPOSITION3.3 (vertical contact angle). If Cm er‡R

e ‡r‡ jgj_L1(BR), then

v(x)ˆ C 

r² jx x0j² q

if x2Br(x0) M otherwise

8<

:

is a supersolution of (6)if Br(x0)B_R. Then for M>C, the minimizer of (6)has vertical contact angle with@B_R.

PROOF. We must show that forClarge enough, divg(rF(rv))‡v g0:

A direct computation shows that inBr(x0) we havervˆ x x0

r² jx x₀j² q

thus

rF(rv)ˆ x x₀ e²r²‡(1 e²)jx x0j²

q :

From this we get that

divg(rF(rv))‡v g m

er‡ x x0

e²r²‡(1 e²)jx x₀j²

q x‡C



r² jx x0j² q

jgj_L1(BR)

m

er

jx x0j



e²r²‡(1 e²)jx x₀j²

q jxj ‡C

r jgj_L1(BR)

m

er R

e ‡C r jgj_L1(BR): Thus ifCm

er‡R

e ‡r‡ jgj_L1(BR)thenvis a super-solution.

If M>C, then considering balls of radius r such that @Br\@BR is reduced to a point, by the comparison Proposition 3.2, ifuminimizes (6) thenM>Cvuand thus by Theorem 3.1 we have

"

ru e²‡ jruj²

q n

#

ˆ1 H^{m 1} a:e:on@BR;

which is the vertical contact angle condition. p The interior regularity of minimizers of (6) easily follows by a result of Giaquinta, Modica and Soucek [6].

PROPOSITION3.4. Let g be aC^a(B_R)function then the minimizer of(6) isC^2;a(B_R).

PROOF. By Theorem 3.3 of [6] we have that minimizers of

BVming(BR)

Z

BR

F(Dgu)‡ Z

BR

G(x;u)dg‡ Z

@BR

ju MjgdH^{m 1}

are locally Lipschitz ifG(x;u) verifies the following hypotheses:

@G

@u ‡

@rxG

@u C.

@²G

@u² 0.

Originally we have G(x;u)ˆ1

2ju g(x)j² which does not verifies ex- actly the hypotheses. However if we set G(x;~ u)ˆC(u) g(x)u‡1

2g(x)² whereC(u)ˆ1

2u²ifjuj CandCconvex,C²with linear growth at infinity then G~ verifies the condition mentioned above. The Euler-Lagrange equation verified by the minimizers withG~ instead ofGis

div_g ru e²‡ jruj² q

0 B@

1 CA‡@C

@uˆg:

…7†

Now we can apply Theorem 3.3 of [6] to find that solutions of (7) are locally Lipschitz. Exactly as in Proposition 3.2 the comparison principle holds for this equation and thusM(respectively M) is a supersolution (respectively a subsolution). This implies that ifCMsolutions of (7) are also solutions of (6) which are thus locally Lipschitz. By classical regularity theory for elliptic equations (see [7]) this implies that the solutions are indeed

C^2;a(BR). p

REMARK3.5. This proposition in particular applies forgconvex since convex functions are locally Lipschitz.

Having only interior regularity it is not possible to directly apply the results of Korevaar [10] which need continuity up to the boundary. The idea will be to use a geometric version of Korevaar's argument to get the convexity of the minimizers.

For simplicity, in this part of the proof we focus on the caseeˆ1. By rescaling, the general case ofe6ˆ1 can be easily recovered (the Gaussian measuregis not invariant by this scaling but it does not matter). Consider now the set (see Figure 1)

Uˆ(x;t)2B_R[ M;M]=tu(x) : …8†

The aim is to show thatUis a convex set. First we need to show that U is regular. For this we follow an idea of Giusti (see [8] and [9]) showing that the complement of U is a solution of a certain obstacle problem.

ForFa set of finite perimeter inR^m‡1letP(F) be defined by~ P(F)~ ˆ

Z

@F

g(x)dH^m(x;t):

P~ is thus the perimeter associated to the measurem(x;t)ˆg(x)dxdt. L et nowH(x;t)ˆ(t g(x))g(x) then we have the following:

PROPOSITION3.6. Let E be the complement of U in R^m[ M;M]

then E is a minimizer of P(F)~ ‡

Z

F

H(x;t)dx dt …9†

among all sets containing B^c_R[ M;M]. As a consequence@E isC¹. PROOF. Let us define the field

z(x;t)ˆ

ru 1‡ jruj²

q ; 1 1‡ jruj² q

0 B@

1

CA (x;t)2B_R] M;M[

n^BR(x) (x;t)2@B_R] M;M[

8>

>>

><

>>

>>

:

:

Thenzis aX2vector field inB_R] M;M[ satisfyingjzj_R^m‡1 ˆ1 and [zn^E]ˆ1 wheren^Eis the outward normal toE. Moreover ifzˆ(z⁰;z_m‡1)

Figure 1.

withz⁰2R^m andzm‡12Rthen setting by a slight abuse of notations div_gzˆdiv_gz⁰‡@zm‡1

@t

we have div_gzˆg u. Hence ifFDEB_R] M;M[, ast5u(x) inE, Z

EnF

(div_gz)dmˆ Z

EnF

(g u)dm Z

EnF

(g t)dm

ˆ Z

EnF

Hdxdtˆ Z

E\F

Hdxdt Z

E

Hdxdt:

On the other hand we have:

Z

EnF

(divgz)dmˆ Z

@(EnF)

[zn^EnF]gdH^m(x;t)

ButEnFˆE\F^cand since

n^Eˆn^Fc H^m a:e: on@E\@F andH^mR^m‡1n(E⁽⁰⁾[E⁽¹⁾[@E)

ˆ0,

@(E\F^c)ˆJ_E;F^c[@E\(F^c)⁽¹⁾

[@F^c\E⁽¹⁾ whereJE;F^c ˆx2@E\@F^c=n^Eˆn^Fc

. Moreover we have:

n^EnF ˆ

n^E in @E\(F^c)⁽¹⁾ n^Fc ˆ n^F in @F^c\E⁽¹⁾ n^Eˆ n^F in J_E;F^c 8>

><

>>

: :

From this we find Z

EnF

(divgz)dmˆ

Z

@E\F⁽⁰⁾

gdH^m Z

@F\E⁽¹⁾

n^FzgdH^m‡ Z

JE;Fc

[zn^E]gdH^m

Z

@E\F⁽⁰⁾

gdH^m Z

@F\E⁽¹⁾

gdH^m‡ Z

JE;Fc

[zn^E]gdH^m:

We thus find:

Z

E\F

Hdxdt Z

E

Hdxdt Z

@E\F⁽⁰⁾

gdH^m Z

@F\E⁽¹⁾

gdH^m‡ Z

JE;Fc

[zn^E]gdH^m:

Similarly, studying what happens onFnEwe get:

Z

E\F

Hdxdt Z

F

Hdxdt Z

@F\E⁽⁰⁾

gdH^m Z

@E\F⁽¹⁾

gdH^m‡ Z

JF;Ec

[zn^F]gdH^m:

Summing these two inequalities and using that R

JE;Fc

[zn^E]gdH^mˆ R

JF;Ec

[zn^F]gdH^mwe have:

Z

@F\(E⁽⁰⁾[E⁽¹⁾)

g(x)dH^m(x;t)‡ Z

F

H(x;t)dxdt Z

@E\(F⁽⁰⁾[F⁽¹⁾)

g(x)dH^m(x;t)‡ Z

E

H(x;t)dxdt:

Adding to this equality R

@E\@Fg(x)dH^m(x;t) and using that H^m((A⁽¹⁾[A⁽⁰⁾[@A)^c)ˆ0 for every set of finite perimeterAR^m‡1, we find as desired that

Z

@F

g(x)dH^m(x;t)‡ Z

F

H(x;t)dxdt Z

@E

g(x)dH^m(x;t)‡ Z

E

H(x;t)dxdt:

The regularity of@Efollows from an old paper of Miranda [11, Teorema 2]

which states thatEis smooth in the neighborhood of the obstacle. Sinceuis smooth by Proposition 3.4,Eis smooth everywhere. We point out that in the paper cited above, the results are written for the classical perimeter without curvature terms. However, the argument is based on a blow-up procedure under which our functional reduces to the classical perimeter. p

We can now prove the convexity ofU.

PROPOSITION3.7. The set U is convex thus u is also convex.

PROOF. Let us define for everyzˆ(x;t)2BR[ M;M] the vertical distance ofztoUby

d^v(z;U)ˆinf…jt t⁰j=(x;t⁰)2U†:

The functiond^vis continuous since@Uis aC¹surface by Proposition 3.6.

U is a compact set thus the function

C(l;z;z⁰)ˆd^v(lz‡(1 l)z⁰;U) for (l;z;z⁰)2[0;1]UU attains its maximum. If this maximum is zero thenUis convex and we are done. Assume on the contrary that this maximum is positive.

By the vertical contact angle condition we can assume that this maximum is attained at pointszandz⁰in the interior ofBR[ M;M]. Moreover, if zˆ(x;t)2U, by decreasing t (which increases C), we can assume that tˆu(x). Analogously we can assume thatz⁰ˆ(x⁰;u(x⁰)). Then we find

C(l;z;z⁰)ˆu(lx‡(1 l)x⁰) lu(x) (1 l)u(x⁰):

We are thus in the situation of applying Korevaar's concavity maximum principle [10, Theorem 1.3] to conclude. We briefly recall the argument for the reader's convenience.

As (l;z;z⁰) is a point of maximum, the gradient inxand inx⁰is zero and thus ru(lx‡(1 l)x⁰)ˆ ru(x)ˆ ru(x⁰):

As the second derivative of C(l;(x‡t;u(x‡t));(x⁰‡t;u(x⁰‡t))) is nonpositive in zero for every directiont2R^mwe get

D²u(lx‡(1 l)x⁰) lD²u(x) (1 l)D²u(x⁰)0:

…10†

Let x⁰⁰ˆlx‡(1 l)x⁰ then u satisfies at the points x, x⁰ and x⁰⁰ the equation

Tr(D²F(ru)D²u)‡ rux‡uˆg where we remind thatF(p)ˆ 

1‡ jpj² q

. SinceFis a convex function,D²F is a positive symmetric matrix and sinceru(x)ˆ ru(x⁰)ˆ ru(x⁰⁰), using (10) we get

C(l;x;x⁰)ˆu(x⁰⁰) lu(x⁰) (1 l)u(x⁰⁰)g(x⁰⁰) lg(x⁰) (1 l)g(x⁰⁰)0

which gives a contradiction. p

We now finally turn to the proof of our main result:

THEOREM 3.8. Let g2L²_g(R^m) be a convex function and u be the minimizer of

BVgmin\L²g(R^m)

Z

R^m

jDguj ‡1 2

Z

R^m

ju gj²dg;

then u is a convex function.

PROOF. By Proposition 3.3 we see that ifuRis the minimizer of (6) then it is convex. Arguing as in [5, Theorem 4.1], we see thatuRconverges locally uniformly toue, the minimizer of (5), whenRgoes to infinity. Analogously, we can letegoes to zero and get thatu_econverges locally uniformly touthe

solution of (4) which is thus convex. p

Let us also notice that along the same lines we can prove the following result:

THEOREM 3.9. Let g be a convex L²_g(R^m) function; then the mini- mizer of

u2Hmin¹_g(R^m)

Z

R^m

jruj² 2 dg‡1

2 Z

R^m

ju gj²dg

is convex.

PROOF. L et Jl(u)ˆl²

Z

R^m



1‡jruj² l² s

1 2

4

3 5dg‡1

2ju gj²dg;

thenu_lminimizesJ_lif and only if it minimizes Z

R^m



l²‡ jruj² q

dg‡ 1 2l

Z

R^m

ju gj²dg:

Thusu_lis convex. Using that for anyp,

l!1liml²



1‡jpj² l² s

1 2

4

3 5ˆjpj²

2 ;

we get the conclusion. p

REMARK3.10. When trying to extend the previous method for more general functionals, a difficulty arise due to the lack of boundary regularity of the minimizers. More precisely, when reasoning as in Proposition 3.6, these functionals give rise to anisotropic perimeters, for which it is not known if the minimizers of the corresponding obstacle problem are smooth in a neighborhood of the obstacle.

Acknowledgements. I warmly thank Matteo Novaga for the numerous discussions we had on this problem and for his constant support. I would also like to acknowledge the hospitality of the Scuola Normale Superiore di Pisa where this work has been done. I finally would like to thank the anonymous referee for his (or her) careful reading of the manuscript and for the useful suggestions.

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Manoscritto pervenuto in redazione il 25 Ottobre 2011.