On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization

On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization

MICHELEMARINI(*) - BERARDORUFFINI(**)

ABSTRACT- We solve a class of weighted isoperimetric problems of the form min

Z

@E

we^Vdx: Z

E

e^Vdxˆconstant 8<

:

9=

;

where wand V are suitable functions on R^d. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 26D10, 35J15, 26D20.

KEYWORDS. Weighted isoperimetric inequalities, symmetrizations, rearrangements.

1. Introduction

In the celebrated paper [12], G. Talenti established several comparison results between the solutions of the Poisson equation with Dirichlet boundary condition (with suitable dataf andE):

Duˆf inE; uˆ0 on@E …1:1†

and the solutions of the corresponding problem wheref andEare replaced by their spherical rearrangements (see [10, Chapter 3] for the definition and main properties of spherical rearrangement). Precisely, he proves that if we denote byv the solution of the problem with symmetrized data, then the

(*) Indirizzo dell'A.: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy.

E-mail: [email protected]

(**) Indirizzo dell'A.: Institut Fourier, 100 rue des maths, BP 74, 38402 St Martin d'HeÁres cedex, France.

E-mail: [email protected]

The author was partially supported by the PRIN 2010-2011 ``Calculus of Varia- tions''.

rearrangementuof the (unique) solutionuof (1.1) is pointwise bounded by v. Moreover he shows that theL^qnorm ofruis bounded, as well, by theL^q norm ofrv, forq2(0;2]. The proof of these facts basically relies on two ingredients: the Hardy-Littlewood-Sobolev inequality and the isoperimetric inequality (see [1] and [10] for comprehensive accounts on the subjects).

Later on, following such a scheme, many other works have been de- veloped to prove analogous comparison results related to the solutions of PDEs involving different kind of operators, see for instance [2], [3], [6], [7], [8], [9], [13] and the references therein. A recurring idea in these works is, roughly speaking, the following: the operator considered is usually linked to a sort of weighted perimeter. Thus initially it is necessary to solve a corresponding isoperimetric problem; then the desired comparison results can be obtained following the ideas contained in [12].

For example in [3] the authors consider a class of weighted perimeters of the form

Pw(E)ˆ Z

@E

w(jxj)dH^{d 1}(x);

whereEis a set with Lipschitz boundary andw:R![0;1) a non-negative function, and prove, under suitable convexity assumptions on the weightw, that the ball centered at the origin is the unique solution of the mixed iso- perimetric problem

minfPw(E):jEj ˆconstantg

wherej jdenotes thed-dimensional Lebesgue measure. As a consequence they prove comparison results, analogous to those considered by Talenti in [12], for the solutions of

div(w²ru)ˆf in E; uˆ0 on@E:

Recently in [4], L. Brasco, G. De Philippis and the second author proved a quantitative version of the weighted isoperimetric inequality considered in [2]. Their proof is achieved by means of a sort ofcalibration technique.

One advantage of this technique is that it is adaptable to other kind of problems, as that of considering other kind of functions in the weighted perimeter (e.g. Wulff-type weights, see [5]), or that of considering dif- ferent measured spaces, as R^d endowed with the Gauss measure.

In this paper we consider degenerate elliptic equations with Dirichlet boundary condition of the form

div(w²e^Vru)ˆf e^V inE; uˆ0 on@E …1:2†

where wandV are two given functions, and we aim to prove analogous

comparison results as those in [12]. The particular form in which is written the measuree^V is due to the later applications, whose main examples are Gauss-type measures, that isV(x)ˆ cjxj². Bearing in mind this instance, we consider a class of mixed isoperimetric problems of the form

min P_we^V(E): Z

E

e^V ˆconstant 8<

:

9=

;

and prove, by means of a calibration technique reminiscent of that devel- oped in [4], that the solutions, under suitable assumptions onVandw, are half-spaces, see Proposition 3.1 and Theorem 3.6. Then, using a suitable concept of rearrangement related to the measures considered, we prove, in the Main Theorem in Section 4, comparison results between the solutions of (1.2) and the solutions of the same equation with rearranged data.

2. Preliminaries on rearrangement inequalities

In this section we introduce the main definitions and properties about the concept of symmetrization and rearrangement we shall make use of.

Let m be a finite Radon measure onR^d, aright rearrangement with respect tomis defined, for any Borel setA, as

R^m_Aˆ f(x₁;x⁰)2RR^{d 1} : x₁>t_Ag;

where t_Aˆinf

t :m(A)ˆm(f(x₁;x⁰)2RR^{d 1} : x₁>tg) . Notice that ifdmˆfdx, for some positive and measurable functionf, then the value oft is uniquely determined.

Given a non-negative Borel function f :R^d![0;‡1), we call right increasing rearrangementoff the functionf^mgiven by

f^m(x)ˆ Z^‡1

0

x_R^m

ff>tg(x)dt

wherex_Ais the characteristic function of the setA. As an aside we notice that the right increasing rearrangement of the characteristic function of a Borel setAcoincides with the characteristic function ofR^m_A. Clearlyf^mis non-negative, increasing with respect to the first variablex₁, and constant on the sets f(x₁;x⁰)2RR^{d 1}:x₁ˆtg, for t2R. Moreover f and f^m share the same distribution function:

m_f(t):ˆm(ff >tg)ˆm(ff^m>tg)ˆm_fm(t):

We furthermore define f^?m:R^‡!R^‡ as the smallest decreasing function satisfyingf^?m(m_f(t))t; in other words

f^?m(s)ˆinfft>0 : m_f(t)<sg:

It is useful to bear in mind that fs:f^?m(s)>tg ˆ[0;m_f(t)] so that by the Layer-Cake Representation Theorem (see for instance [10]) we have

Z

m(fx1>tg)

0

f^?m(s)dsˆ Z¹

t

m_f(s)dsˆ Z

fx1>tg

f^m(x)dx:

…2:1†

We conclude this section by proving theHardy-Littlewoodrearrange- ment inequality related to the right symmetrization.

LEMMA2.1 (Hardy-Littlewood rearrangement inequality). Let f and g be non-negative Borel functions from R^d to R. Then for any non-negative Borel measuremwe have

Z

R^d

f g dm Z

R^d

f^mg^mdm:

PROOF. We have Z

R^d

f g dmˆ Z

R^d

Z¹

0

Z¹

0

x_ff>tg(x)x_fg>sg(x)dt ds dm(x)

ˆ Z¹

0

Z¹

0

Z

R^d

x_ff>tg\fg>sg(x)dm(x)dt ds

ˆ Z¹

0

Z¹

0

m(ff >tg \ fg>sg)dt ds

Z¹

0

Z¹

0

min(m(ff >tg);m(fg>sg))dt ds

ˆ Z¹

0

Z¹

0

min(m(ff^m>tg);m(fg^m>sg))dt ds

ˆ Z¹

0

Z¹

0

m(ff^m>tg \ fg^m>sg)dt dsˆ Z

R^d

f^mg^mdm;

where we used the fact thatff^m>tgandfg^m>sgare half-spaces of the formf(x1;x⁰)2RR^{d 1}:x1>rgfor somer2Rand so

min(m(ff^m>tg);m(fg^m>sg))ˆm(ff^m>tg \ fg^m>sg):

p REMARK2.2. Settinggˆx_Ain Lemma 2.1and thanks to(2.1)we get

Z

A

f dx Z

R^m_A

f^m(x)dxˆ Z^m(A)

0

f^?m(s)ds:

…2:2†

3. A class of weighted isoperimetric inequalities

Given a measurable functionV:R^d!Rwe denote bym[V] the abso- lutely continuous measure whose density equals e^V, that is, for any mea- surable setER^d

m[V](E)ˆ Z

E

e^V(x)dx;

in what follows with the scope of simplifying the notation, and if there is no risk of confusion, we will drop the dependence of V, writingminstead of m[V]. Moreover we will often adopt the notationxˆ(x1;x⁰)2RR^{d 1}and denote byR_Ainstead ofR^m[V]_A the right rearrangement ofAwith respect to the measurem[V]. Given a Borelweightfunctionw:R![0;‡1] we de- fine, for any open setAwith Lipschitz boundary, the following concept of weighted perimeter:

Pw;V(A)ˆ Z

@A

w(x1)e^V(x)dH^{d 1}(x):

In the following proposition we show that, under suitable conditions onw andV, the half-spaces of the formf(x₁;x⁰):x₁>tgare the only minimizers of the weighted perimeter among the sets of fixed volume with respect to the measurem[V].

PROPOSITION3.1. Let AR^dbe a set with Lipschitz boundary. Sup- pose that w:R!R^‡and V:R^d!Rare C¹-regular functions satisfying the following assumptions:

…i† m(A)ˆm(RA)<‡ 1,

…ii† the function @1V(x) depends only on x1 and g(x):ˆ w⁰(x1) w(x1)@1V(x)is a non-negative decreasing function on the real line.

Then

P_w;V(A)P_w;V(R_A):

…3:1†

PROOF. We start by noticing that ifPw;V(A)ˆ ‡ 1there is nothing to prove. Hence we can suppose that

P_w;V(A)<‡ 1:

…3:2†

Lete₁ˆ(1;0;. . .;0)2R^d and consider the vector field e₁w(x₁)e^V(x). Its divergence is given by

div( e1w(x1)e^V(x))ˆ( w⁰(x1) w(x1)@1V(x))e^V(x)ˆg(x)e^V(x): By an application of the Divergence Theorem we have

Z

A

g(x)dm(x)ˆ Z

A

div( e1w(x1)e^V(x))dx

ˆ Z

@A

w(x₁)e^V(x)hn_A(x); e₁idH^{d 1}(x)

Z

@A

w(x₁)e^V(x)dH^{d 1}(x)ˆP_w;V(A);

…3:3†

wherenA(x) is the outer unit normal to@Aatx. LettAbe a real number such that the right half-space R_Aˆ f(x₁;x⁰):x₁ t_Ag satisfies m(R_A)ˆm(A).

Then, since the outer normal ofR_A is the constant vector field e₁, the inequality in (3.3) turns into an equality if we replaceAwithR_A. Notice that by condition (ii) and (3.3) we have

P_w;V(R_A)ˆ Z

RAnA

g dm‡ Z

RA\A

g dmg(t_A)m(A)‡P_w;V(A):

Thanks to assumption (i) and (3.2) such quantities are finite and so we get P_w;V(A) P_w;V(R_A)

Z

A

g(x)dm(x) Z

RA

g(x)dm(x):

Since, by definition,m(A)ˆm(R_A)<‡ 1again by condition (i) we obtain

m(AnRA)ˆm(RAnA)<‡ 1. Thus Z

A

g(x)dm(x) Z

RA

g(x)dm(x)ˆ Z

AnRA

g(x)dm(x) Z

RAnA

g(x)dm(x)

ˆ Z

AnRA

(g(x) g(t_Ae₁))dm(x) Z

RAnA

(g(x) g(t_Ae₁))dm(x):

…3:4†

Since everyx2AnR_A(respectivelyx2R_AnA) satisfieshx;e1i<t_A(re- spectivelyhx;e₁i>t_A), by condition…ii†we deduce

P_w;V(A) P_w;V(R_A) Z

AnRA

jg(x) g(t_Ae₁)jdm(x)‡ Z

RAnA

jg(x) g(t_Ae₁)jdm(x)

ˆ Z

ADRA

jg(x) g(tAe1)jdm0;

…3:5†

where ADR_Aˆ(AnR_A)[(R_AnA) stands for the symmetric difference

betweenAandR_A. This concludes the proof. p

REMARK 3.2 (Necessity of the assumptions). We stress that the in- tegrability condition (i) is necessary to formulas (3.3) and (3.4) (and thus to our proof) to work.

Concerning condition (ii), we note that it is needed just for technical reasons. Nonetheless we stress that our proof offers a slightly stronger inequality than (3.1). Indeed the right-hand side of (3.5) may be seen as a modulus of continuity of theL¹distance betweenAandRA. Thus it would be interesting to understand how much our hypotheses are far from op- timality (compare also with [4, Remark 2:3]).

REMARK 3.3 (Equality cases). An inspection of the proof of Propo- sition 3.1, and in particular of inequality (3.3), shows that if w>0, then we have equality in (3.1) only ifAis equal to the half spaceRA, up to set of zero d-dimensional Lebesgue measure. On the other hand, if the set fwˆ0ghas positive Lebesgue measure, we can not expect any kind of uniqueness for the equality cases of such an inequality.

EXAMPLE3.4. A non-trivial example fulfilling condition (ii) of Proposi- tion 3.1 is the following

V(x₁;x⁰)ˆ c(x₁jx₁j ‡ jx⁰j²); w(x₁)ˆe ^ax1;

witha;c>0 constants satisfyinga² 2c0. To prove this fact we initially observe that ifx16ˆ0 such a condition is equivalent to require that

w⁰⁰(x1)‡V₁⁰⁰(x1)w(x1)‡V₁⁰(x1)w⁰(x1)0 …3:6†

which turns out to be equivalent, in our example, to a² 2c‡2acjx₁j 0:

Then, since w⁰(x₁) w(x₁)@₁V(x₁) is continuous inx₁ˆ0, condition (ii) is satisfied everywhere.

To transform inequality (3.1) into a well posed isoperimetric problem, it would be more advisable to eliminate the integrability hypothesis (i) in Proposition 3.1 by requiring that the measure m(R^d)<‡ 1. This fact, together with ordinary differential inequality required in assumption (ii), is seldom satisfied.

Hence, to get other instances of functions which fulfill inequality (3.6) together with the integrability property…i†of Proposition 3.1 it is worth restricting our attention to the half-space

R^d_‡ˆ f(x1;x⁰)2RR^{d 1}:x1 >0g:

As an immediate corollary of Proposition 3.1 we get that the solution of the problem

min

P_w;V(A):AR^d_‡; m(A)ˆc; @ALipschitz …3:7†

is given byRc ˆ fx1tcgwheretcis such thatm(Rc)ˆc.

REMARK3.5. Notice that the non-mixed Gauss case, wconstant and V(x)ˆ cjxj², is not covered by our hypotheses. Nevertheless in this case examples of functionswwhich satisfy the hypotheses of Proposition 3.1 are given byw(t)ˆt ^a with a1 orw(t)ˆb‡e ^at, with a;b0 such that a² 2c(1‡b)>0 (as can be easily seen reasoning as in the previous ex- ample). In the latter case at least ifbˆ0 we have that

we^V ˆe^a2=(4c)exp c x‡e1 a

2c ²

;

wheree₁ˆ(1;0;. . .;0)2R^d, which can be rephrased (¹) as the fact that the

(¹) As suggested us by an anonymous Referee.

solutions of the isoperimetric problem in the half-spaceR^d_‡with (suitable) mixed Gaussian conditions

min

P_gs;h(E):g_s;0(E)ˆconstant; ER^d_‡; @ELipschitz

are right-half spaces. Here we denoted byg_s;hthe normal distribution whose covariance matrix issId and whose mean vectorhis given byhˆ a

2ce₁. We recall that, as pointed out in the Introduction, similar problems related to the Gauss measure are considered in [2], [6], [8], [9] and [13].

Notice that we defined the perimeterP_w;V only for sets with Lipschitz boundary, but for our later applications it will be useful to have a definition of perimeter which comprehends also less regular subsets of R^d. A mea- surable setAis said to have locally finite (Euclidean) perimeter (we refer to [11] for a complete overview on the subject) if there exists a vector- valued Radon measuren_AcalledGauss-Green measureof the setAsuch that, for everyT2C_c¹(R^d;R^d), it holds true that

Z

A

divTˆ Z

R^d

hT;dn_Ai:

The perimeter ofAis defined in terms of the total variation of the Gauss- Green measure of A as P(A)ˆ jn_Aj(R^d). For any set A of locally finite perimeter we then define theweighted perimeter P_w;V by

P_w;V(A)ˆwe^Vjn_Aj(R^d):

Since when Ahas Lipschitz boundaryjn_Aj ˆ H^{d 1} @A, the above defi- nition is coherent with the one given at the beginning of this section on such sets.

THEOREM 3.6. Let w and V non-negative and C¹-regular functions satisfying condition (ii) of Proposition 3.1. Suppose moreover that m(R^d_‡)<‡ 1; then the problem

min

P_w;V(A):AR^d_‡; m(A)ˆc

admits a solution, and this solution coincides with the one of(3.7).

PROOF. LetAbe a measurable set of locally finite perimeter and sup- pose, by contraddiction, thatP_w;V(A)<P_w;V(R_A). We start by noticing that

Pw;V(RA)<‡ 1, indeed, recalling (3.3) we have that P_w;V(R_A)ˆ

Z

RA

g(x)dm(x)g(0)m(A):

By [11, Theorem II.2.8] we can find a sequence of setsAnwith smooth boundary such thatx_An!x_A inL¹_loc(R^d) andjn_Anj* jn_Aj, where * in- dicates the weak*convergence of Radon measures. Since m(R^d_‡)<‡ 1, we also have that

x_An!x_A in L¹(R^d;m) …3:8†

and, sincewe^Vis a continuous function

n!1lim Z

R^d

we^Vdjn_Anj ˆ Z

R^d

we^Vdjn_Aj:

…3:9†

Thanks to (3.9) and Proposition 3.1 we get P_w;V(A)ˆ lim

n!1P_w;V(A_n) lim

n!1P_w;V(R_An):

We are left to show that lim

n!1P_w;V(R_An)ˆP_w;V(R_A), but jP_w;V(R_A) P_w;V(R_An)j g(0)jm(A) m(A_n)j;

and we can conclude thanks to (3.8) and the fact thatm(R^d_‡)<‡ 1. p

4. Main result

In this section we consider sets ER^d_‡ and we define dmˆe^Vdx, R_Eˆ fx₁>t_Egwhere t_E2Ris such thatm(R_E)ˆm(E) andfˆf^m the right rearrangement of a functionf with respect tom. In what follows we consider problems of the form

div(w²e^Vru)ˆf e^V inE

uˆ0 on@E

…4:1†

which must be intended in weak sense. Precisely, a solution of (4.1) is a functionu2H¹₀(e^V;w²e^V;E), defined as the closure ofC₀¹(E) with respect to the norm

kuk_H1

0(e^V;w²e^V;E)ˆ Z

E

u²e^Vdx‡ Z

E

jruj²we^Vdx 0

@

1 A

1=2

;

and which satisfies Z

E

hru;rfiw²e^Vdxˆ Z

E

ffe^Vdx …4:2†

for anyf2H¹₀(e^V;w²e^V;E).

The main scope of this section is to prove a prioriestimates for the solutions of problem (4.1). For this reason we shall always consider that a solutionuexists. Clearly this requirement depends on the choice ofw,V and f. General instances of such functions for which the existence of a solution for problem (4.1) is guaranteed, can be found in [14] (see also [2], [8], [9], and [13]). Here we limit ourselves to state that most of the examples considered in Remark 3.5, as the mixed-Gaussian case V(x)ˆ cjxj², w(t)ˆb‡e ^at witha² 2c(1‡b)>0 andbstrictly positive, are covered by the cases considered in [14], wheneverf 2L²(E;e^V).

MAINTHEOREM. Suppose that the set ER^d_‡ˆ f(x₁;x⁰):x₁>0gand the functions w:[0;‡1]!(0;‡1] and V :R^d!R satisfy the hy- potheses of Proposition3.1. Consider the two problems

div(w²e^Vru)ˆf e^V in E

uˆ0 on@E

( …4:3†

and

div(w²e^Vrv)ˆfe^V in R_E

vˆ0 on@R_E

…4:4†

where 0<f 2L²(R^d_‡;m). Then the problem (4.4)has as solution the one variable function v(z)given by

v((z;z⁰))ˆv(z)ˆ Z

m(RE)

m(fx1zg)

1 h²(s)

Z^s

0

f(j)dj 0

@

1 Ads;

…4:5†

where

h(m)ˆw(F ¹(m)) Z

R^{d 1}

m(F ¹(m);x⁰)dx⁰; …4:6†

beingF(t)ˆm(fx1>tg). Moreover, for any solution u of the problem(4.3),

we have

u(x)v(x);

…4:7†

and, for any q2(0;2], Z

E

jruj^qw^qdm Z

RE

jrvj^qw^qdm …4:8†

PROOF. Let us suppose for the moment that the functionvgiven in (4.5) is a solution for the problem (4.4). To prove (4.7) and (4.8) we consider the functionsf_hdefined as

f_h(x)ˆ

sign (u) if juj>t‡h u(x) tsign u(x)

h if juj 2[t;t‡h)

0 if juj<t;

8>

>>

><

>>

>>

:

where 0t<ess supjujandh>0. Notice that, for everyh>0,f_h is an admissible test function, since the solution u belongs to the space H¹₀(e^V;w²e^V;E). Then (4.2) turns into

1 h

Z

fjuj2[t;t‡h)g

hru;ruiw²dmˆ1 h

Z

fjuj2[t;t‡h)g

f(u t u juj)dm‡

Z

fjuj>t‡hg

fsign (u)dm:

Taking the limit forh!0, we get d

dt Z

fjuj>tg

jruj²w²dmˆ Z

fjuj>tg

f dm:

…4:9†

Let us analyze the left-hand side of equation (4.9). We claim that the fol- lowing inequality holds true for almost everyt:

d dt

Z

fjuj>tg

jruj²w²dm

dtd

R

fjuj>tgjrujw dm

₂ m⁰_u(t) ; …4:10†

wherem_u(t) is the distribution function ofuintroduced in the Section 2.

Indeedm_u(t) is a decreasing function and thence it is derivable for almost everyt, thanks to the HoÈlder inequality we get

d dt

Z

fjuj>tg

jrujw dmˆlim

h!0

1 h

Z

t<juj<t‡h

jrujw dm

lim

h!0

Z

ft<juj<t‡hg

jruj²w²dm 0

B@

1 CA

1=2 Z

ft<juj<t‡hg

1 h² dm 0

B@

1 CA

1=2

ˆlim

h!0

1 h

Z

ft<juj<t‡hg

jruj²w²dm 0

B@

1 CA

1=2

1 h

Z

ft<juj<t‡hg

1dm 0

B@

1 CA

1=2

ˆ d

dt Z

fjuj>tg

jruj²w²dm 0

B@

1 CA

1=2

m⁰_u(t)₁₌₂

By the Co-Area formula and the fact that wis strictly positive andC¹- regular, we easily get that the set fu>tg is a set of locally finite (Euclidean) perimeter. Thus, thanks to Proposition 3.1 and Theorem 3.6 we get

…4:11† d dt

Z

fjuj>tg

jrujw dmˆ Z

fjuj ˆtg

w dmˆPw;V(fjuj>tg)Pw;V(fu>tg):

We introduce the function

F(t)ˆm(fx₁>tg):

…4:12†

We recall that the weight functionwis constant on the boundary of the super level sets of u, so that the perimeter offu>tg can be written as

P_w;V(fu>tg)ˆw(t) Z

R^{d 1}

m(t;x⁰)dx⁰:

Moreovert2Rsatisfiesm_u(t)ˆF(t) that istˆF ¹(m_u(t)) (notice thatF is a strictly decreasing function and thus invertible) so that we can write the previous formula as

…4:13† P_w;V(fu>tg)ˆw(F ¹(m_u(t))) Z

R^{d 1}

m(F ¹(m_u(t));x⁰)dx⁰:ˆh(m_u(t)):

Plugging (4.12) in (4.10), and recalling (4.13) we get that d

dt Z

fjuj>tg

jruj²w²dmh(m_u(t))² m⁰_u(t) : …4:14†

We pass now to estimate the right-hand side of (4.9): equation (2.2) with Aˆ fjuj>tgturns into

Z

fjuj>tg

f dm Z

fjuj>tg

fdmˆ Z

m_u(t)

0

f^?(s)ds:

…4:15†

Combining (4.15) and (4.14) we get R

mu(t) 0 f^?(s)ds

! m⁰_u(t) h²(m_u(t)) 1:

…4:16†

Reasoning analogously for the function v, we easily see that, since v is constant on every setfx1ˆtg and since vˆv, (4.16) holds for v as an equality. Consider now the real function

F(r)ˆ R^r

0 f(s)ds h(r)² ;

and letGbe a primitive ofF. SinceF0, we have thatGis increasing.

Moreover by our previous analysis we have that F(m_u(t))m⁰_u(t) 1ˆF(m_v(t))m⁰_v(t):

We recall that herem⁰_u(t) denotes the derivative almost everywhere of the functionm_u(t). Moreovert7!G(m_u(t)) is a monotone non-increasing func- tion which satisfies the chain rule in any point of differentiability ofm_u, so that, by [1, Corollary 3:29], we get that

G(m_u(t))G(m_u(0))‡ Z^t

0

F(m_u(t))m⁰_u(t)dt:

…4:17†

On the other hand, beingm_v(t) an absolutely continuous function (sincevis a C¹-regular with positive derivative one variable function) we have

G(m_v(t))ˆG(m_u(0))‡ Z^t

0

F(m_v(t))m⁰_v(t)dt;

…4:18†

so that, since G(m_v(0))ˆG(m_u(0)), we get that G(m_u(t))G(m_v(t)).

This implies that m_u(t)m_v(t) for any t and hence that uv, since u and v depends only on x1 and are increasing functions of such a variable.

We pass now to the proof of (4.8). Using the HoÈlder inequality and reasoning as before we obtain, for 0<q2,

d dt

Z

fjuj>tg

jruj^qw^qdmˆlim

h!0

1 h

Z

ft<juj<t‡hg

jruj^qw^qdm

lim

h!0

1 h

Z

ft<juj<t‡hg

jruj²w²dm 0

B@

1 CA

q=2

1 h

Z

ft<juj<t‡hg

dm 0

B@

1 CA

1 q=2

ˆ d

dt Z

fjuj>tg

jruj²w²dm 0

B@

1 CA

q=2

( m⁰_u(t))^{1 q=2}:

Recalling (4.9) and (4.15) we have d

dt Z

fjuj>tg

jruj²w²dm Z

m_u(t)

0

f(s)ds;

thus

d dt

Z

fjuj>tg

jruj^qw^qdm Z

m_u(t)

0

f^?(s)ds 0

B@

1 CA

q=2

( m⁰_u(t))^{1 q=2}: …4:19†

Combining (4.19) and (4.16) we finally get d

dt Z

fjuj>tg

jruj^qw^qdm( m⁰_u(t)) h(m_u(t)) ¹ Z

mu(t)

0

f^?(s)ds 0

B@

1 CA

q

:

By integrating on both side between 0 and‡ 1, we get Z

E

jruj^qw^qdm Z¹

0

( m⁰_u(t)) h(m_u(t)) ¹ Z

mu(t)

0

f^?(s)ds 0

B@

1 CA

q

dt:

We perform the change of variablesrˆm_u(t), so that the above equation turns into

Z

E

jruj^qw^qdm Z^m(E)

0

h(r) ¹ Z^r

0

f^?(s)ds 0

@

1 A

q

dr:

By a straightforward inspection of those steps we notice thatvsatisfies Z

RE

jrvj^qw^qdmˆ Z¹

0

( m⁰_v(t)) h(m_v(t)) ¹ Z^mv(t)

0

f^?(s)ds 0

B@

1 CA

q

dt;

By performing the change of variablesrˆm_v(t) we find Z

RE

jrvj^qw^qdmˆ Z

m(RE) 0

h(r) ¹ Z^r

0

f^?(s)ds 0

@

1 A

q

dr:

Sincem(E)ˆm(R_E) we get the desired result.

We are left to prove that the functionv given by (4.5) is a solution of problem (4.4). We start by noticing that equation (4.16) suggests how to derive (4.5): indeed, as we pointed out, any solution v of (4.4) such that vˆvsatisfies

R

m_v(t) 0 f^?(s)ds

h²(m_v(t)) m⁰_v(t)ˆ 1:

By integrating both sides between 0 andrwe obtain Z^r

0

R

m_v(t) 0 f^?(s)ds

h²(m_v(t)) m⁰_v(t)dtˆ r:

so that, by performing the change of variablesmˆm_v(t), we get Z

m(RE)

mv(r)

R^m

0 f^?(s)ds

h²(m) dmˆr

which is equivalent to

v(z;z⁰)ˆ Z

m(RE)

mfx1>zg

R^m

0 f^?(s)ds h²(m) dm;

that is (4.5). Notice that vis strictly decreasing and belongs to C^1;1_loc(R_E).

Indeed, recalling (4.6) one can explicitly compute

rv(z;z⁰)ˆe₁@v

@z(z;z⁰)ˆ e₁ R

mfx1>zg 0 f^?(s)ds w²(z) R

R^{d 1}

e^V(z;x0)dx⁰;

wheree₁ˆ(1;0;. . .;0)2R^d. Sincef^?is a decreasing and locally integrable function, thenf^?2L¹_loc(R); thus, beingz7!m(fx1>zg)C¹-regular, we get that ^mfxR^1>zg

0 f^?(s)dsis a locally Lipschitz function. Moreover the denomi- nator is locally Lipschitz as well, and locally bounded away from zero.

Hence we have that rv is locally Lipschitz. Thus, recalling that@1V de- pends only on the first variablex1 it is possible to explicitly compute the divergence ofw²rve^V and check that it satisfies (4.4). This concludes the proof of the theorem.

Acknowledgments. The authors thank L. Brasco and G. De Philippis for useful discussions on the topic. They are also grateful to the anonymous referee for several suggestions and remarks.

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Manoscritto pervenuto in redazione il 16 dicembre 2013.