In this article we prove a Logarithmic Sobolev type inequality and a Poincar´e type inequality for functions in the anisotropic Gaussian Sobolev space

Electronic Journal of Differential Equations, Vol. 2019 (2019), No. 89, pp. 1–15.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

ANISOTROPIC LOGARITHMIC SOBOLEV INEQUALITY WITH A GAUSSIAN WEIGHT AND ITS APPLICATIONS

FILOMENA FEO, GABRIELLA PADERNI

Abstract. In this article we prove a Logarithmic Sobolev type inequality and a Poincar´e type inequality for functions in the anisotropic Gaussian Sobolev space. As an application we study a class of equations, whose anisotropic elliptic condition is given in term of the density of Gauss measure. Finally some extensions of the main results are given for a class of weighted (not Gaussian one) anisotropic Sobolev spaces.

1. Introduction

In the previous years anisotropic problems and spaces have been extensively studied by many authors, motivated by their applications to the mathematical modeling of physical and mechanical processes in anisotropic continuous medium.

LetN ≥2 and 1≤p1, . . . , pN <+∞. Roughly speaking an anisotropic Sobolev space is a space of functions u such that i-th partial derivative of u belongs to the Lebesgue space L^pi with some exponent pi. If Ω is a bounded open set of R^N with Lipschitz continuous boundary a Sobolev type inequality (see e.g. [24] for Sobolev type inequalities in the Lebesgue spaces and [23] for similar inequalities in the Lorentz spaces) holds for functions belonging to the anisotropic Sobolev space W₀^1,−→p(Ω), defined as the closure ofC₀^∞(Ω) with respect to normPN

i=1k∂xiukLpi(Ω). Indeed (see e.g. [24]) there exists a constantCS such that

kukL^q(Ω)≤C_S

N

X

i=1

k∂xiukLpi(Ω) ∀u∈W₀^1,−→p(Ω), (1.1) whereq=p^∗= _N−p^{N p} ifp < N orq∈[1,+∞[ ifp≥N andpdenote the harmonic mean, i.e. ¹_p =_N¹ PN

i=1 1

pi. Moreover an anisotropic Poincar´e inequality holds (see e.g. [15]):

kukLpi(Ω)≤p_i

2c(Ω)k∂xiukLpi(Ω) ∀u∈W₀^1,−→p(Ω)∀i= 1, . . . , N, (1.2) where c(Ω) = sup_x,y∈Ω(x−y,ei), {e1, . . . ,eN} is the canonical basis of R^N and (·,·) denotes the standard scalar product in R^N. When p < N, inequality (1.1) implies the continuous embedding ofW₀^1,−→p(Ω) intoL^q(Ω) for everyq∈[1, p^∗]. On

2010Mathematics Subject Classification. 46E35, 35A12, 46E30.

Key words and phrases. Gauss measure; logarithmic Sobolev-Poincar´e inequality;

anisotropic spaces.

c

2019 Texas State University.

Submitted February 7, 2019. Published July 17, 2019.

1

the other hand the continuity of the embeddingW₀^1,→−p(Ω)⊂L^pmax(Ω) withp_max:=

max{p1, . . . , p_N} relies on inequality (1.7). Then p_∞ := max{p^∗, p_max} turns out to be the critical exponent: there is a continuous embeddingW₀^1,→−p(Ω)⊂L^q(Ω) for q∈[1, p_∞].

Also there is an increasing interest to Sobolev type inequalities involving weighted Sobolev spaces. In this article we take into account the Gaussian weight. In this contest Gross [17] proved (see [14] for more comments and references) the inequality

Z

R^N

|u|^plog|u|dγ≤ p 2 Z

R^N

|∇u|²|u|^p−2dγ+kuk^p_Lp(R^N,γ)logkuk_Lp(R^N,γ) (1.3) for u ∈ W^1,p(R^N, γ) with 1 < p < +∞, where γ states for the Gauss measure.

Unlike the classical Sobolev inequality it is independent on dimension and easily extends to the infinite dimensional case. In terms of functional spaces inequal- ity (1.3) implies the imbedding of weighted Sobolev space W^1,p(R^N, γ) into the weighted Zygmund spaceL^p(logL)^1/2(R^N, γ) (see§2.2 for definition). The imbed- ding holds forp= 1 as well and it is related to Gaussian isoperimetric inequality.

Forp= 2 Gross inequality (1.3) entails [9] that

ku−uγk^∗_L2(logL)^1/2(R^N,γ)≤Ck∇uk_Lp(R^N,γ), whereuγ :=R

R^Nu(x)γ and then

kuk^∗_L2(logL)^1/2(R^N,γ)≤ C(k∇uk_L2(R^N,γ)+kuk_L2(R^N,γ)) for some constant independent on the dimension. Herek·k^∗_L2(logL)1/2(

R^N,γ)states for the rearrangement invariant quasinorm in the Zygmund spaceL²(logL)^1/2(R^N, γ) (see§2.2 for definition). This kind of inequalities hold for 1≤p <+∞as well (see e.g. [9, 20]).

A Logarithmic Sobolev-Poincar´e inequality (see [13]) is proved in for functions in the weighted Sobolev space W₀^1,p(Ω, γ), defined as the closure of C₀^∞(Ω) with respect to normk∇uk_Lp(Ω,γ). Let 1≤p <+∞and Ω be an open subset ofR^N (not necessary bounded) withγ(Ω)<1 andu∈W₀^1,p(Ω, γ), thenu∈L^p(logL)^1/2(Ω, γ) and

kuk^∗_Lp(logL)^1/2(Ω,γ)≤Ck∇ukL^p(Ω,γ) (1.4) for some constantCdepending only onpandγ(Ω) (see [13]). Using the continuous embedding between Zygmund spaces the previous inequality yields the following Poincar´e inequality:

kuk_Lp(Ω,γ)≤ Ck∇uk_Lp(Ω,γ) ∀u∈W₀^1,p(Ω, γ) (1.5) for some positive constantC(independent of u, but depending on γ(Ω) and p). If γ(Ω) = 1 inequality (1.4) holds if in the right hand-side we take into account the norm ofuas well. We recall that the analogue inequality whenu∈W^1,p(Ω, γ) is studied in [13] and, as one can expect, smoothness assumption on ∂Ω has to be required.

In this article we consider functions such thati-th partial derivative ofubelongs to some weighted (with respect to Gauss measure) Lebesgue space L^pi for some exponent p_i. More precisely we prove a Logarithmic Sobolev inequality type in- equality and a Poincar´e type inequality for functions belonging to the anisotropic

Gaussian Sobolev spaceW₀^1,−→p(Ω, γ) (see §2 for the definition). As in the Gauss- ian isotropic case we have to require some additional hypothesis on Ω. Instead of γ(Ω)<1 we assume that Ω is an open subset ofR^N such that

(H1) γ₁(a_i, b_i)<1 with a_i = inf_x∈Ω(x,e_i) and b_i = sup_x∈Ω(x,e_i). for a fixed i= 1, . . . , N.

Theorem 1.1. Let N ≥2 andΩbe an open subset ofR^N with Lipschitz boundary such that (H1)holds for a fixed i= 1, . . . , N. Then for anyu∈W₀^1,−→p(Ω, γ),

Z

Ω

|u(x)|^pilog^pi/2(2 +|u(x)|)ϕ(x)dx1/pi

≤c1(pi)h1 2

1− 1

log(γ₁(a_i, b_i)) i1/2

∂u

∂x_i

_Lpi(Ω,γ),

(1.6)

wherec1(pi)is a constant depending only onpi. Moreover an anisotropic Poincar´e inequality holds.

Theorem 1.2. Let N ≥2 andΩbe an open subset ofR^N with Lipschitz boundary such that (H1)holds for a fixed i= 1, . . . , N. Then for anyu∈W₀^1,−→p(Ω, γ),

kuk_Lpi(Ω,γ)≤c₂(p_i)

− 1

2 log(γ1(ai, bi)) ^1/2

k∂_xiuk_Lpi(Ω,γ), (1.7) wherec2(pi)is a constant depending only onpi.

The constants in (1.6) and (1.7) depend on the 1-dimensional Gauss measure of the diameter of Ω in i-th direction as it happens in (1.2), where the Lebesgue measure of the diameter is involved. In order to emphasize that a 1-dimensional measure is considered we used the notationγ₁.

We stress that hypothesis (H1) guarantees thata_iorb_iis finite. Let us consider examples of sets for which the previous theorem can be applied. If Ω = {x ∈ R^N : x₁ > ω} or Ω = {x ∈ R^N : x₁ < ω} for some ω ∈ R, then the previous theorems hold for i= 1 but not fori= 2, . . . , N. Instead if we take into account Ω ={x∈R^N :xi > ωi fori= 1, . . . , N} for someω1, . . . , ωN ∈R, Theorems 1.1 and 1.2 holds for anyi= 1, . . . , N.

Corollary 1.3. Let N≥2andΩbe an open subset ofR^N with Lipschitz boundary such that (H1)is in force for anyi= 1, . . . , N. Then for anyu∈W₀^1,−→p(Ω, γ)

Z

Ω

|u(x)|^pmaxlog^pmax2 (2 +|u(x)|)ϕ(x)dx_pmax¹

≤c(pmax)h1 2

1− 1

log(γ1(amax, bmax))

i^1/2X^N

i=1

∂u

∂xi

_Lpi(Ω,γ),

(1.8)

wherepmax= max{p1, . . . , pN}=pjfor somej∈ {1, . . . , N},amax=aj,bmax=bj

andc(pmax)is a constant depending only onpmax.

An anisotropic Sobolev-Poincar´e inequality with Orlicz target norm reads as follows:

kuk_Lpmax(logL)^1/2 ≤c(p_max)h1 2

1− 1

log(γ1(amax, bmax))

i1/2 N

X

i=1

∂u

∂xi

_Lpi(Ω,γ),

where kuk_Lpmax(logL)^1/2 is the Orlicz norm of uin L^pmax(logL)^1/2(Ω, γ) (see (2.5) for a definition). As consequence the continuous embedding ofW₀^1,−→p(Ω, γ) into the Zygmund spaceL^pmax(logL)^1/2(Ω, γ) is proved.

This embedding and the previous results are no longer true if the zero trace condition on the boundary is removed as in the isotropic case with weight or not.

Moreover we stress that if (H1) is not in force inequality (1.6) holds if in the right hand-side we take into account the norm ofuas well.

This article is organized as follows. In Section 2 we recall some definitions and properties. The main results are proved in Section 3. Section 4 is dedicated to the application of these inequalities to study a class of equations, whose anisotropic elliptic condition is given in term of the density of Gauss measure. In particular we prove some uniqueness results. Finally in the last section we show some gener- alization of our main results to a class of weighted (not Gaussian one) anisotropic Sobolev spaces.

2. Preliminaries

2.1. Gauss measure and Gaussian rearrangements. Let γ be theN-dimen- sional Gauss measure onR^N defined by

dγ:=ϕ(x)dx:= (2π)^−N/2exp

−|x|² 2

dx, x∈R^N normalized byγ(R^N) = 1.

LetN ≥2 and Ω be an open subset ofR^N not necessary bounded with Lipschitz boundary and let 1≤p1, . . . , pN <∞beN real numbers. A measurable functionu belongs toL^pi(Ω, γ) ifR

Ω|u|^pidγ <+∞. The anisotropic Gaussian Sobolev space (see e.g. [24] for the definition without weight) is defined as

W^1,−→p(Ω, γ) ={u∈W^1,1(Ω, γ) :ux_i ∈L^pi(Ω, γ) fori= 1, . . . , N}

and is a Banach space with respect to the norm kuk_W1,−→p(Ω,γ)=kuk_L1(Ω,γ)+

N

X

i=1

kux_ik_Lpi(Ω,γ). (2.1) As usual the space W₀^1,−→p(Ω, γ) is defined as the closure ofC₀^∞(Ω) with respect to the norm (2.1). When inequality (1.7) is in force, the norm (2.1) is equivalent to the norm that involves only the partial derivatives.

It is well known that an isoperimetric inequality with respect to Gauss measure (see e.g. [10]) holds. For all subsetsE⊂R^N it follows thatP(E)≥ϕ(Φ⁻¹(γ(E))), where Φ(τ) is the Gauss measure of the half-space

x∈R^N :x1> τ for every τ∈R∪ {−∞}andP(E) is the perimeter with respect to Gauss measure ofE. We recall that the isoperimetric functionIγ(t) has the following asymptotic behavior

I_γ(t) :=ϕ(Φ⁻¹(t))∼t 2 log1

t ^1/2

fort→0⁺; 1⁻. (2.2) Following [10] we define the one dimensional Gaussian decreasing rearrangement of uby

u^~(s) = inf{t≥0 :γ({x∈Ω :|u(x)|> t})≤s} s∈]0,1],

and the Gaussian rearrangement ofubyu^?(x) =u^~(Φ(x1)) for x∈Ω^?, where Ω^?={x= (x₁, . . . , x_N)∈R^N :x₁> ω}

is the half-space such thatγ(Ω^?) =γ(Ω). A Polya-Sz¨ego type inequality (see [22]) holds for 1≤p <+∞:

k∇u^?kL^p(Ω^?,γ)≤ k∇ukL^p(Ω,γ) ∀u∈W₀^1,p(Ω, γ). (2.3) 2.2. Zygmund spaces. We say that a measurableubelongs to the Zygmund space (see e.g. [3])L^r(logL)^α(Ω, γ) for 1≤r <+∞andα∈Rif the quantity

kuk^∗_Lr(logL)^α(Ω,γ):=Z γ(Ω) 0

[(1−logt)^αu^~(t)]^rdt^1/r

(2.4) is finite. We emphasize that the Zygmund space L^r(logL)^α(Ω, γ) coincides with the Lebesgue spaceL^r(Ω, γ) whenα= 0. If 1< r < p <∞and−∞< α, β <∞ we obtainL^p(logL)^α(Ω, γ),→L^r(logL)^β(Ω, γ). It is clear from (2.4) that the space L^r(logL)^α(ϕ,Ω) decreases asαincreases. We remark that (2.4) is a quasinorm and is equivalent to the norm obtained replacingu^~(t) withu^~~(t) :=¹_tRt

0u^~(s)dsfor p >1. Moreoveru,u^~ andu^? have the same Zygmund quasinorm.

If we consider the Zygmund space L^r(logL)^α(Ω, γ) as an Orlicz space, a mea- surable functionubelongs to it if and only if [|u(x)|log^α(2 +|u(x)|)]^r is integrable with respect toγ. Moreover its Orlicz norm is defined as

kuk_Lr(logL)^α(Ω,γ):= infn λ >0 :

Z

Ω

u(x) λ

log^α(2 +

u(x) λ

)r

dγ≤1o

(2.5) and it is not in general equivalent to the quasinorm (2.4). The following inequality is useful in working with Zygmund spaces.

Proposition 2.1. Suppose r > 0,1 ≤ q < +∞ and −∞ < α <+∞. Let ψ be a nonnegative measurable function on (0, b) with 0 < b ≤ 1, then the following inequality holds:

Z 1

0

t^r(1−logt)^α Z 1

t

ψ(s)ds^qdt t

^1/q

≤cZ 1 0

(t^1+r(1−logt)^αψ(t))^qdt t

^1/q

(2.6) with a constantc=c(r, q, α)are independent on ψ and onb.

Whenb = 1 the previous inequality is proved in [3]. The proof works for 0 <

b <1 as well and it is easy to check that the constant does not depend onb.

3. Proofs of main results

The idea is to estimate u as a function of xi variable using the Logarithmic Sobolev-Poincar´e inequality with respect to Gauss measure in dimension 1. To do this we need an explicit dependence of the involved constant with respect to the data. For convenience of the reader we detail the dependence of such a constant.

3.1. Comments on logarithmic Sobolev-Poincar´e inequality(1.4). The main aim of this subsection is to obtain an explicit dependence of the constant with re- spect to the domain Ω in (1.4). The proof of this inequality is based on properties of rearrangements of functions and on asymptotic behaviour Gaussian isoperimetric function.

Letu∈W₀^1,p(Ω, γ). First we observe that (1−logt)≤

1− 1

log(γ(Ω)) log1

t for 0< t < γ(Ω). (3.1)

Using (2.4), (2.6), (2.2), (3.1) and (2.3) we obtain kuk^∗_Lp(logL)^1/2(Ω,γ)

^p

≤c(p) Z γ(Ω)

0

t(1−logt)^1/2 d dtu^~(t)

p

dt

≤c(p)h1 2

1− 1

log(γ(Ω))

i^p/2Z γ(Ω)

0

h d dtu^~(t)

ϕ1(Φ⁻¹(t))i^p dt

=c(p)h1 2

1− 1

log(γ(Ω)) ip/2

k∇u^?k^p_Lp(Ω^?,γ)

≤c(p)h1 2

1− 1

log(γ(Ω)) ip/2

k∇uk^p_Lp(Ω,γ)

for some positive constantc(p)depending onp, that can be vary from line to line, yielding

kuk^∗_Lp(logL)^1/2(Ω,γ)

^p

≤c(p)h1 2

1− 1

log(γ(Ω)) i^p/2

k∇uk^p_Lp(Ω,γ), (3.2) i.e. (1.4) with an explicit dependence of the constant on the set Ω.

An easy consequence of (3.2) is the inequality Z

Ω

|u(x)|^plog^p/2(2 +|u(x)|)ϕ(x)dx

≤c3(p)h1

2(1− 1

log(γ(Ω)))ip/2

k∇uk^p_Lp(Ω,γ)

(3.3)

for some positive constantc₃(p) depending onp. Indeed observing that by proper- ties of rearrangements of functions it follows that

u^~(t)≤u^~~(t) := 1 t

Z t

0

u^~(s)ds≤kuk_L1(Ω,γ)

t and using (3.2) we obtain

Z

Ω

|u(x)|^plog^p/2(2 +|u(x)|)ϕ(x)dx

≤ Z γ(Ω)

0

u^~(t) log^1/2(2 +u^~(t))^p dt

≤ Z γ(Ω)

0

[u^~(t) log^1/2

2 + kuk_L1(Ω,γ)

t

]^pdt

≤c(p) Z γ(Ω)

0

u^~(t)(1−logt)^1/2p dt :=c(p) kuk^∗_Lp(logL)^1/2(Ω,γ)

p

≤c(p)h1 2

1− 1

log(γ(Ω)) ip/2

k∇uk^p_Lp(Ω,γ).

wherec(p) is a positive constant depending onp, that can be vary from line to line.

3.2. From 1-dimensional to anisotropic logarithmic Sobolev-Poincar´e in- equality. By denseness it is sufficient to prove (1.6) foru∈C₀¹(Ω). Let us fixi∈ {1, . . . , N}such thatγ1(ai, bi)<1 withai= infx∈Ω(x,ei) andbi= sup_x∈Ω(x,ei).

This hypothesis guarantees that the section of Ω in the direction e_i is not all the line a.e.. We assume without loss of generality thata_i ∈R and b_i ∈ R∪ {+∞}.

It follows that Ω ⊆ {x ∈ R^N : a_i < x_i < b_i}. We consider u as defined on the whole R^N, setting to 0 outside suppt(u). For for all x∈ R^N, we set x= (x_i, x⁰) in order to emphasize its i-th component. Since the Gauss measure is a product measure we can writeϕ(x) =ϕ_N−1(x⁰)ϕ1(xi), whereϕN−1 andϕ1are the density of the (N−1)-dimensional Gauss measureγN−1and 1-dimensional Gauss measure γ1respectively. For any fixedx⁰∈R^N−1, by (3.3) and properties of rearrangement of functions we obtain

Z b_i

ai

|u(xi, x⁰)|^pilog^pi/2(2 +|u(xi, x⁰)|)ϕ1(xi)dxi

≤c(pi)h1 2

1− 1

log(γ(a_i, b_i))

ipi/2Z b_i

ai

|∂x_iu(xi, x⁰)|^piϕ1(xi)dxi

(3.4)

for some constantc(pi) depending only on pi. Now multiplying by ϕN−1(x⁰) and integrating onR^N−1, we obtain

Z

Ω

|u(x)|^pilog^pi/2(2 +|u(x)|)ϕ(x)dx

≤c(pi)h1 2

1− 1

log(γ(ai, bi))

i^pi/2Z

Ω

∂

∂xi

u(x)

pi

ϕ(x)dx,

which is (1.6).

3.3. From 1-D logarithmic Sobolev-Poincar´e inequality to anisotropic Poincar´e inequality. By denseness it is sufficient to prove (1.7) for u∈C₀¹(Ω).

We consider the same notations and assumptions of the previous subsection. For any fixed x⁰ ∈ R^N−1, by (3.2) and properties of rearrangement of functions we obtain

Z b_i

a_i

|u(xi, x⁰)|^piϕ1(xi)dxi

=

Z γ(ai,bi)

0

|u^~(t_i, x⁰)|^pidt

≤ sup

0<t<γ1(ai,bi)

(1−logt)^−pi2

Z γ1(ai,bi)

0

u^~(t, x⁰)(1−logt)^1/2pi

dt

:= sup

0<t<γ₁(a_i,b_i)

(1−logt)^−pi2ku(·, x⁰)k_Lpi(logL)^1/2((a_i,b_i),γ₁)

≤c(pi)h1 2

1− 1

log(γ1(ai, bi)) i^pi/2

1−log(γ1(ai, bi))−pi/2

× Z bi

a_i

∂

∂xi

u(x_i, x⁰)

p_i

ϕ₁(x_i)dx_i,

where u^~(t, x⁰) is one dimensional Gaussian decreasing rearrangement of u with respect to x_i, for each x⁰ fixed. Now multiplying by ϕ_N−1(x⁰) and integrating on

R^N−1, we obtain Z

Ω

|u(x)|^piϕ(x)dx≤c(p_i)

− 1

2 log(γ(a_i, b_i)) ^pi/2Z

Ω

∂

∂x_iu(x)

p_i

ϕ(x)dx_i, for some constantc(pi) depending only on pi; thus inequality (1.7) is proved.

Remark 3.1. A similar computation proves that the constant in (1.5) is given by c(p)[−2 log(γ(Ω))¹ ]^1/2for some constantc(p) depending only onp.

4. Application to PDEs

Let us consider the class of nonlinear homogeneous Dirichlet problems

−

N

X

i=1

∂xiai(x, u,∇u) =F in Ω u= 0 on∂Ω,

(4.1)

where N ≥2, Ω is an open subset of R^N with Lipschitz boundary such that (H1) holds for every i = 1, . . . , N, ai : Ω×R×R^N → R is a Carath´eodory function fulfilling the degenerated anisotropic ellipticity condition

N

X

i=1

a_i(x, s, ξ)ξ_i≥λ

N

X

i=1

|ξ_i|^piϕ(x) ∀s∈R, ξ∈R^N a.e. x∈Ω, (4.2) with 1< pi<∞andλ >0 and the growth condition

|ai(x, s, ξ)| ≤[ν₁|s|^pi−1+ν₂|ξi|^pi−1]ϕ(x) ∀s∈R, ξ∈R^N a.e. x∈Ω, (4.3) withν1≥0 andν2>0 andF is an element of dual space. For example the datum can be given byf ϕ−PN

i=1(giϕ)x_iwithf ∈L^p0max(logL)⁻¹²(Ω, γ) andgi∈L^p0i(Ω, γ), where p⁰_i states for the H¨older conjugate exponent of pi. We observe that the equation in (4.1) is related to Ornstein-Uhlenbeck operator in the isotropic case.

We take into account weak solutions to problem (4.1). The natural space for searching them is the weighted anisotropic Sobolev spaceW₀^1,−→p(Ω, γ). When the datum F is in the dual space a weak solution to problem (4.1) is a function u∈ W₀^1,−→p(Ω, γ) such that

Z

Ω

a_i(x, u,∇u) ∂

∂xi

ψ dx=hF, ψi ∀ψ∈W₀^1,→−p(Ω, γ), (4.4) where h·,·i is the duality pairing. We stress that under the assumptions (4.2)- (4.3) every term in (4.4) is well-defined and the operator −PN

i=1∂x_iai(x, u,∇u) is monotone and coercive on the weighted anisotropic Sobolev space W₀^1,−→p(Ω, γ).

Then there exists (see e.g. [19]) at least a weak solutionu∈W₀^1,−→p(Ω, γ) to problem (4.1).

In what follows we are interested in some uniqueness results. As in the classical case, to guarantee uniqueness the main hypotheses are a strongly monotonicity and a Lipschitz continuity of the involved operator. More precisely we suppose that every functionai satisfies the following strongly monotonicity condition

(a_i(x, s, ξ)−a_i(x, s, ξ⁰))(ξ_i−ξ_i⁰)≥α(ε+|ξ_i|+|ξ_i⁰|)^pi−2|ξ_i−ξ⁰_i|² (4.5) withα >0 andε≥0 and the following locally Lipschitz continuity

|ai(x, s, ξ)−a_i(x, s⁰, ξ)| ≤β|ξi|^pi−1|s−s⁰| (4.6)

with β >0. We stress Lipschitz continuity condition (4.6) is necessary to get the uniqueness of a solution ( for a counterexample see e.g. [4] in the weighted case and [7] in a bounded domain whenϕ(x)≡1).

As in no weighted case (see e.g. [2]) we are able to prove an uniqueness result whenε= 0 in (4.5) and at least onep_i≤2.

Theorem 4.1. Let us suppose that there exists j ∈ {1, . . . , N} such that pj ≤ 2 and that (4.5)withε= 0and (4.6)hold. Then problem (4.1)has at most one weak solution inW₀^1,→−p(Ω, γ).

As in the no weighted case if all pi >2 for everyi = 1, . . . , N we have to take into account only the caseε >0 in ((4.5). Indeed otherwise the uniqueness is not guaranteed as for thep−Laplace operator (for a counterexample see e.g. [4] in the weighted case and [1] in a bounded domain whenϕ(x)≡1).

Theorem 4.2. Let us assume that p_i >2 for every i = 1, . . . , N and that (4.5) with ε >0 and (4.6) hold. Then problem (4.1) has at most one weak solution in W₀^1,−→p(Ω, γ).

The proofs of Theorems 4.1 and 4.2 follow the idea in [1] (see also [4] for isotropic weighted case).

Let us consider the case when the datum isf ϕwith f ∈L^p0max(logL)⁻¹²(Ω, γ).

The restrictionε >0 when allpi>2 can be avoided iffdoes not change sign (see [8]

and [12] in the no weighted case and [4] for isotropic weighted case). We obtain the following uniqueness result holding for the model operatorPN

i=1(|∂x_iu|^pi−2∂x_iu ϕ)x_i

with allpi>2, which does not fulfill (4.5) withε >0.

Theorem 4.3. Let us assume that pi >2 for every i = 1, . . . , N and that (4.3) with ν1 = 0,(4.5)with ε= 0and (4.6) hold and that the sign off is constant on Ω. Then problem (4.1)has at most one weak solution inW₀^1,−→p(Ω, γ).

Whenϕ(x) ≡1 and Ω is bounded, uniqueness results for elliptic problems are proved for example in [1, 5, 8, 12, 16, 21] (see also the bibliography therein).

4.1. Proof of Theorem 4.1. Letuandvbe two weak solutions to problem (4.1).

Let (u−v)⁺ := max{0, u−v}, D = {x ∈ Ω : (u−v)⁺ > 0}, Dt = {x ∈ D : (u−v)⁺ < t} for t ∈ [0,sup(u−v)⁺[ and let us suppose that D has positive measure. LetTt(s) be the truncation function at levelt, i.e.

Tt(s) = min{t,max{s,−t}}. (4.7)

Taking^Tt((u−v)_t ⁺⁾as test function in (4.4) written foruandv, making the difference of the two equations, we obtain

N

X

i=1

Z

D_t

[ai(x, u,∇u)−ai(x, v,∇v)]∂xiψ dx≤0.

Hypothesis (4.5) withε= 0 and (4.6) yield

N

X

i=1

Z

D_t

|∂_xiψ|²(|∂_xiu|+|∂_xiv|)^pi−2dγ≤ β α

N

X

i=1

Z

D_t

|∂_xiv|^pi−1|∂_xiψ|dγ. (4.8)

By Young inequality with someδ >0 we obtain Z

Dt

|∂x_iv|^pi−1|∂x_iψ|dγ

≤δ 2

Z

D_t

|∂x_iψ|²(|∂x_iu|+|∂x_iv|)^pi−2dγ+ 1 4δ

Z

D_t

(|∂x_iv|+|∂x_iv|)^pidγ.

(4.9)

Putting (4.9) in (4.8) and choosingδsmall enough we obtain

N

X

i=1

Z

D_t

|∂_xiψ|²(|∂_xiu|+|∂_xiv|)^pi−2dγ≤c₁

N

X

i=1

Z

D_t

(|∂_xiv|+|∂_xiv|)^pidγ (4.10) for some positive constant c1 independent on t. Let pj ≤ 2. Using Poincar´e inequality (1.7), Young inequality and (4.10) we obtain

γ(D\Dt) = Z

Dt

|ψ|dγ≤C Z

Dt

|∂x_iψ|dγ

≤ C 2

hZ

D_t

|∂_xjψ|²

(|∂_xju|dγ+|∂_xjv|)^2−pj dγ+ Z

D_t

(|∂xjv|+|∂xjv|)^2−pjdγi

≤ c1C 2

N

X

i=1

Z

Dt

(|∂x_iu|+|∂x_iv|)^pidγ+C 2

Z

Dt

(|∂x_ju|+|∂x_jv|)^2−pjdγ

=: Λ(t),

where C is the constant in Poincar´e inequality (1.7). Since limt→0Λ(t) = 0 we obtain

γ(D) = lim

t→0γ(D\Dt) = 0 (4.11)

and the conclusion follows.

Remark 4.4. In the last step of proof of Theorem (4.1) we only need that Poincar´e inequality (1.7) holds forj∈ {1, . . . , N} such thatpj<2. Then that we can relax the hypothesis on Ω, requiring that (H1) is fulfilled for such a indexj.

4.2. Proof of Theorem 4.2. Arguing as in the proof of Theorem 4.1 we obtain

N

X

i=1

Z

D_t

(ε+|∂xiu|+|∂xiv|)^pi−2|∂xiψ|²dγ≤β α

N

X

i=1

Z

D_t

|∂xiv|^pi−1|∂xiψ|dγ (4.12) Let us estimate the right hand side. By Young inequality with someδ >0 we have

Z

D_t

|∂x_iv|^p−1|∂x_iψ|dγ≤ 1 4δ

Z

D_t

|∂x_iv|^pidγ+δ 2

Z

D_t

|∂x_iv|^pi−2|∂x_iψ|²dγ. (4.13) Choosingδsmall enough inequalities (4.13) and (4.12) yield

N

X

i=1

Z

D_t

(ε+|∂_xiu|+|∂_xiv|)^p−2|∂_xiψ|²dγ≤c₁

N

X

i=1

Z

D_t

|∂_xiv|^pidγ:= Λ(t) (4.14) for some constantc1independent oft. Moreover Young inequality and (4.14) imply

Z

D

|∂x_iψ|dγ= Z

Dt

|∂x_iψ|dγ

≤1

2γ(D_t) +1 2

Z

D_t

|∂_xiψ|²dγ≤γ(D_t)

2 + Λ(t) 2ε^pi−2.

(4.15)

On the other hand Poincar´e inequality (1.7) gives γ(D\Dt) =

Z

D\Dt

ψ dγ≤ Z

D

ψ dγ ≤C Z

D

|∂x_iψ|dγ.

Observing that lim_t→0Λ(t) = 0, by (4.15) we obtain (4.11) and we have the con- clusion.

4.3. Proof of Theorem 4.3. Letuandv be two weak solutions to problem (4.1) and let (u−v)⁺ := max{0, u−v}, D = {x∈Ω : (u−v)⁺>0}, Dt = {x∈ D : (u−v)⁺< t}fort >0 and let us suppose thatDhas positive measure. We proceed by steps.

Step 1. We prove that

t→0lim 1 t²

N

X

i=1

Z

D_t

(|∂_xiu|+|∂_xiv|)^pi−2|∂_xi(u−v)|²dγ= 0. (4.16) Fort >0, denoting byTtthe function defined as in (4.7), putting as test function Tt[(u−v)⁺] in (4.4) we obtain

N

X

i=1

Z

Ω

[a_i(x, u,∇u)−a_i(x, v,∇v)]∂_xiT_t[(u−v)⁺]dx= 0.

DenotingD¹_i ={x∈Dt,|∂x_iu| ≤ |∂x_iv|}and D_i² ={x∈Dt,|∂x_iv| ≤ |∂x_iu|}, we obtain

N

X

i=1

Z

D¹_i

[ai(x, v,∇u)−ai(x, v,∇v)]∂x_i(u−v)dx

+

N

X

i=1

Z

D_i²

[ai(x, u,∇u)−ai(x, u,∇v)]∂x_i(u−v)dx

≤ −

N

X

i=1

Z

D_i¹

[a_i(x, u,∇u)−a_i(x, v,∇u)]∂xi(u−v)dx

−

N

X

i=1

Z

D_i²

[ai(x, u,∇v)−ai(x, v,∇v)]∂x_i(u−v)dx for everyt >0. By (4.5) and (4.6) we have

α

N

X

i=1

Z

Dt

(|∂x_iu|+|∂x_iv|)^pi−2|∂x_i(u−v)|²dγ

≤β

N

X

i=1

Z

D¹_i

|∂xiu|^pi−1|∂xi(u−v)| |u−v|dγ

+β

N

X

i=1

Z

D_i²

|∂x_iv|^pi−1|∂x_i(u−v)| |u−v|dγ

≤βt

N

X

i=1

Z

D_t

[min{|∂_xiu|,|∂_xiv|}]^pi−1|∂_xi(u−v)|dγ.

(4.17)

Using Young’s inequality we obtain α

N

X

i=1

Z

D_t

(|∂xiu|+|∂xiv|)^pi−2|∂xi(u−v)|²dγ

≤βt

N

X

i=1

Z

Dt

(|∂x_iu|+|∂x_iv|)^pi−1|∂x_i(u−v)|dγ

≤α 2

N

X

i=1

Z

D_t

(|∂xiu|+|∂xi|)^pi−2|∂xi(u−v)|²dγ

+β²t² 2α

N

X

i=1

Z

Dt

(|∂x_iu|+|∂x_iv|)^pidγ and then

α 2t²

N

X

i=1

Z

Dt

(|∂x_iu|+|∂x_iv|)^pi−2|∂x_i(u−v)|²dγ

≤ β² 2α

N

X

i=1

Z

D_t

(|∂_xiu|+|∂_xiv|)^pidγ.

Since the second term in the previous estimate approaches zero as t → 0, (4.16) follows.

Step 2. We prove that

N

X

i=1

Z

D

ai(x, u,∇u)∂x_iΨdx= lim

t→0

Z

Ω

fTt[(u−v)⁺]

t Ψdγ

N

X

i=1

Z

D

a_i(x, v,∇v)∂xiΨdx= lim

t→0

Z

Ω

fT_t[(u−v)⁺]

t Ψdγ

(4.18)

for every Ψ∈L^∞(Ω)∩W^1,−→p(Ω, γ).

Taking ^Tt[(u−v)_t ^+]Ψ as test function in (4.4), we obtain

N

X

i=1

Z

Ω

a_i(x, u,∇u)∂_xiΨT_t[(u−v)⁺]

t dx+1

t

N

X

i=1

Z

D_t

a_i(x, u,∇u)∂_xi(u−v)Ψdx

= Z

Ω

fT_t[(u−v)⁺]

t Ψdγ.

We easily pass to the limit in the first term by using Lebesgue dominated conver- gence theorem. For the second term using (4.3) and H¨older inequality we obtain

1 t

Z

D_t

ai(x, u,∇u)∂x_i(u−v)Ψdx

≤ν2kΨk_L∞(Ω)

1 t²

Z

Dt

(|∂x_iu|+|∂x_iv|)^pi−2|∂x_i(u−v)|²dγ1/2

×Z

D_t

(|∂xiu|+|∂xiv|)^pidγ^1/2 ,

which tends to zero by (4.16). Then we obtain (4.18).

Step 3. We prove that D has zero measure. Taking Ψ = 1 in (4.18) we obtain

t→0lim Z

Ω

fT_t[(u−v)⁺]

t dγ= 0.

Since the sign off is constant we obtainf χ_D= 0 a.e. in Ω and the right-hand side of (4.18) is zero.

Now takingψ=T_k(u) in (4.18) and passing to the limit ask→ ∞, we obtain

N

X

i=1

Z

D

a_i(x, u,∇u)∂xiu dx= 0.

By (4.5) withε= 0 and (4.3) withν1= 0 we obtain

N

X

i=1

Z

D

|∂x_iu|^pidγ= 0.

Then

∂_xiu= 0 a.e. onD for everyi∈ {1, . . . , N}. (4.19) By (4.17) and (4.19) it follows that ∂x_iv = 0 a.e. on Dt for every t > 0 and for every i ∈ {1, . . . , N} and then in D. Then ∂x_i(u−v) = 0 a.e. on D for every i∈ {1, . . . , N}. Sinceu=v= 0 on∂Ω, by Poincar´e inequality (1.7)

Z

D

|u−v|^pidγ= Z

Ω

|(u−v)⁺|^pidγ≤C Z

D

|∂x_i(u−v)|^pidγ= 0.

Then the conclusion follows.

Remark 4.5. In the last step of proofs of Theorems (4.2) and (4.3) we only need that Poincar´e inequality (1.7) holds for somej∈ {1, . . . , N}, that means that we can relax the hypothesis on Ω, requiring that (H1) is fulfilled for somej∈ {1, . . . , N}.

5. An extension to Boltzmann measures Let us consider the weight

Z⁻¹exp(−W(x))∈L¹(Rⁿ), (5.1) where W(x) is a positive smooth function. We can see (5.1) as the density of the following measure onR^N, called Boltzmann measure, defined by

dµ=Z⁻¹exp(−W(x))dx x∈R^N

and normalized byµ(R^N) = 1. Let us suppose that measureµsatisfies an isoperi- metric inequality and its isoperimetric functionIµ(t) is estimated by the Gaussian isoperimetric function:

Iµ(t)≤cµIγ(t) (5.2)

for a suitable positive constantc_µ>0. For example (5.2) is satisfied if the Hessian matrix satisfies

D²W(x)≥c²_µId (5.3)

as symmetric matrix uniformly inx(see [18]). We remark that the Gauss measureγ is such a measure.

Under the previous assumption arguing as in subsection 3.1 it is possible to prove that

kuk^◦_Lp(logL)^1/2(Ω,µ)

^p

≤c_µc(p)h1 2

1− 1 log(µ(Ω))

i^p/2

k∇uk^p_Lp(Ω,µ) (5.4)

for some constant c(p) depending only on p, and then the analogue of our main results. The main tools are a suitable definition of rearrangement and Polya-Sz¨ego inequality for it. Ifuis a measurable function in Ω, we define

u^◦(s) = inf{t≥0 :µ({x∈Ω :|u|> t})≤s} fors∈]0,1]

the one dimensional rearrangement of uwith respect to Boltzmann measure and u(x) =u^◦(Φ(x1)) forx∈Ωthe rearrangement with respect to Boltzmann, where Ω is the half-space such that γ(Ω) = µ(Ω). In [11] is proved the Polya-Sz¨ego inequality

k∇uk_Lp(Ω,γ)≤√

cµk∇uk_Lp(Ω,µ). (5.5) In this frameworku^◦,u and (5.5) play the role ofu^~, u^?and (2.3) respectively.

Moreoverkuk^◦_{Lp(logL)1/2(Ω,µ)} is defined as in (2.4) withu^~ replaced byu^◦.

Stating from (5.4) arguing as in §3.2 and §3.3 we can prove the analogue of Theorem 1.1, Theorem 1.2, and Corollary 1.3. Moreover using such inequalities it is possible to study the analogue of problem (4.1), where the density of Gauss measure is replaced by the (5.1) following the ideas of§4.

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Filomena Feo

Dipartimento di Ingegneria, Universit`a degli Studi di Napoli ‘Pathenope’, Centro Di- rezionale Isola C4 80143 Napoli, Italy

Email address:[email protected]

Gabriella Paderni

Dipartimento di Ingegneria, Universit`a degli Studi di Napoli ‘Pathenope’, Centro Di- rezionale Isola C4 80143 Napoli, Italy

Email address:[email protected]