Optimal limiting embeddings for Δ-reduced Sobolev spaces in <span class="mathjax-formula">$ {L}^{1}$</span>

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Ann. I. H. Poincaré – AN 31 (2014) 217–230

www.elsevier.com/locate/anihpc

Optimal limiting embeddings for -reduced Sobolev spaces in L ¹

Luigi Fontana

, Carlo Morpurgo

aDipartimento di Matematica e Applicazioni, Universitá di Milano-Bicocca, Via Cozzi, 53, 20125 Milano, Italy bDepartment of Mathematics, University of Missouri, Columbia, MO 65211, USA

Received 18 January 2012; received in revised form 13 April 2012; accepted 26 February 2013 Available online 14 March 2013

Abstract

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet prob- lems withL¹data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be considered as limiting cases of the Hansson–Brezis–Wainger spaces, for the optimal embeddings of borderline Sobolev spacesW₀^{k,n/ k}.

©2013 Elsevier Masson SAS. All rights reserved.

1. Introduction

In this paper we are concerned with special kinds of the so-called reduced Sobolev spaces, namely the spaces defined by

W^2,1(Ω)=

u∈W₀^1,1(Ω): u∈L¹(Ω)

(1) and

W_,0^2,1(Ω)=closure ofC₀^∞(Ω)in the normu1 (2)

which we name-reduced Sobolev spaces. HereΩis a bounded open set ofRⁿ, andW₀^k,p(Ω)denotes the closure of the set ofC^∞functions compactly supported inΩ, in the normuk,p=(

|α|kD^αu^pp)^1/p. The spacesW^2,1(Ω) andW_,0^2,1(Ω)could be regarded as natural domains for the Dirichlet Laplacian, as an unbounded operator inL¹(Ω).

Indeed, forf∈L¹(Ω)the problem−u=f has a unique solutionu∈W^2,1(Ω), and ifΩ is smooth enough then suchuis the limit ofC^∞functions inΩwhich are continuous up to the boundary, with 0 boundary value. The same considerations can be made for theL^pversionsW^2,pandW_,0^2,p, obtained by replacingW₀^1,1withW₀^1,pandu1

withupin (1) and (2). There is, however, an important difference: ifp >1 thenW_,0^2,p(Ω)=W₀^2,p(Ω), and (ifΩ is smooth enough)W^2,p(Ω)=W₀^1,p(Ω)∩W^2,p(Ω), but these assertions are both false in the casep=1. The reason for this is, essentially, that theL¹ norms of the second partial derivatives cannot be controlled byu1 (see for example[17]). For a generalL¹theory of second order elliptic equations see[8].

* Corresponding author.

E-mail addresses:luigi.fontana@unimib.it(L. Fontana),morpurgoc@missouri.edu(C. Morpurgo).

0294-1449/$ – see front matter ©2013 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2013.02.007

From the above discussion it should be apparent that such spaces are natural choices if one would like to study the summability properties of solutions of the Dirichlet problem inL¹, or the exceptional casen=2 of the Moser–

Trudinger embeddingW^2,

n 2

0 →e^L

n n−2

.

In a recent paper[12]Cassani, Ruf and Tarsi investigated sharp embedding properties of the-reduced spaces in (1) and (2), for smoothΩ. Among the main results of[12]are the sharp forms of the embeddings ofW^2,1(Ω)into the Zygmund spaceL_exp(Ω), whenn=2, and into the weak-Lⁿⁿ⁻² spaceLⁿ−ⁿ2,∞(Ω), whenn3. These spaces are defined by the quasi-norms

u^∗_Lexp= sup

0<t|Ω|

u^∗(t)

1+log^|Ω_t^|, u^∗n

n−2,∞= sup

0<t|Ω|tⁿ⁻ⁿ²u^∗(t), (3)

whereu^∗ denotes the decreasing rearrangement ofuon(0,∞), and the sharp forms of the embeddings derived in [12, Thms. 1, 2]are written as

u^∗_Lexp 1

4πu1, n=2, (4)

u^∗n

n−2,∞ 1

nⁿ⁻ⁿ²(n−2)ω^2/n_n−1u1, n3 (5)

where ω_n−1 denotes the volume of the (n−1)-dimensional unit sphere. The quantities on the left-hand side are quasi-norms defining the spaces L_exp(Ω)and L^n−n2,∞(Ω), respectively; the constants on the right-hand sides are sharp, that is, they cannot be replaced by smaller constants.

In[12]the following slightly better estimate is in fact obtained for anyu∈W^2,1(Ω):

u^∗(t)N_|^∗_Ω|(t)u1, 0< t|Ω|, n2 (6)

where

N_|^∗_Ω|(t)=

⎧⎨

⎩

1

4πlog^|Ω_t^| ifn=2,

1 nⁿ⁻ⁿ²(n−2)ω^2/n_n−1

(t⁻ⁿ⁻ⁿ² − |Ω|⁻ⁿ⁻ⁿ²) ifn3, (7) denotes the decreasing rearrangement of the Green function of the Laplacian for the ball of volume|Ω|, with pole at the origin (see the proofs of Thms. 1, 3 and Prop. 12 in[12]).

It must be noted that inequality (6) was obtained several years ago by Alberico and Ferone (see[3, Theorem 4.1 and Remark 4.1] for the casen=2, and Theorem 5.1 for the casen3, which trivially yields (6)). In[3]it is in fact shown thatu^∗(t)N_|^∗_Ω|(t)P u1, for a general class of second order elliptic operatorsP, such that the Dirichlet problemP u=f admits a unique weak solutionu∈L¹, for eachf ∈L¹(Ω); in such generality, however, one cannot expect the inequality to be sharp. Related results are also contained in[5]and[4].

Regarding the analogous results for the spaceW_,0^2,1(Ω), i.e. the case of compactly supported functions, only partial results were obtained in[12], which however revealed an intriguing aspect: among all functions ofW_,0^2,1(Ω)which are either radial or nonnegative, inequalities (4), (5) and (6) continue to hold but the constants arehalved. In particular, Cassani, Ruf and Tarsi proved that (see[12, proofs of Props. 14 and 16]) for anyt∈(0,|Ω|]

u^∗(t)1

2N_|^∗_Ω|(t)u1, u∈W_,0^2,1(Ω), u0 oruradial, n2 (8) and consequently[12, Thm. 5, Props. 14, 16]

u^∗_Lexp 1

8πu1, n=2, (9)

u^∗n

n−2,∞ 1

2nⁿ⁻ⁿ²(n−2)ω^2/n_n−1u1, n3, (10)

for anyu∈W_,0^2,1(Ω)which is either nonnegative or radial, and with sharp constants, within that class of functions.

One of the original motivations of this work was to find out whether the inequalities in (9), and (10) would still be valid, and therefore sharp, in the whole spaceW_,0^2,1(Ω).

The first main result of this paper is the following sharp version of (8): ifΩ is open and bounded, then for any t∈(0,|Ω|]

u^∗(t)2^−2/nN_|^∗_Ω|(t)u1, u∈W_,0^2,1(Ω), n2. (11) and the constant 2^−2/nin (11) is sharp, in the sense that it cannot be replaced with a smaller constant iftis allowed to be sufficiently small. We will also prove sharpness of (11) for any givent whenΩ is either a ball (n=2) or the whole ofRⁿ (n3). As a consequence of (11) we then find that allowinguto be an arbitrary function inW_,0^2,1(Ω) (not just nonnegative or radial) inequality (9) continues to hold, with sharp constant, whereas (10) is replaced with

u^∗n

n−2,∞ 2^−2/n

nⁿ⁻ⁿ²(n−2)ω^2/n_n−1u1, n3, with sharp constant.

To prove (11), we will first rederive (8) (and also (6)) for arbitrary open and boundedΩ, as a relatively straight- forward consequence of Talenti’s comparison theorem (which was also the starting point in [3] and[12]) and a well-known formula that goes back to Talenti[19], for the solution of the Dirichlet problem on a ball with radial data (see (40), (41)). The use of such formula in combination with Talenti’s type comparison theorems allows one to obtain optimal norm estimates of the solutionuof a Dirichlet problem−u=f in terms of norms off; this idea was already mentioned and used elsewhere (see for example[5, Prop. 3.1]and comments thereafter, and also[3, proof of Theorem 4.1]).

The presence of the factor¹₂in (8) is perhaps better clarified in our proof, which is based on the simple observation that ifuis compactly supported inΩ, then

Ωu=0, and

Ω

(u)⁺dx=

Ω

(u)⁻dx=1

2u1 (12)

where(u)⁺and(u)⁻ denote the positive and negative parts ofu. The proof of (11) will be then obtained by carefully combining estimates for the distribution functions of the positive and the negative parts ofu. We will also introduce natural families of radial extremal functions for (6) and (8), essentially Green’s potentials of normalized characteristic functions of balls or annuli; by suitably translating such functions we will be able to produce a family of extremals for (11).

An immediate consequence of (11) whenn=2 is the following Brezis–Merle type inequality sup

u∈W_,0^2,1(Ω) Ω

e^α|

u(x)|

u1 dx 8π

8π−α|Ω|, α <8π, (13)

where the left-hand side is infinite ifα=8π, and with sharpness of the constant _8π^8π_−α whenΩ is a ball. The same inequality holds forW^2,1(Ω)with 8πreplaced by 4π:

sup

u∈W^2,1(Ω) Ω

e^α|

u(x)|

u1dx 4π

4π−α|Ω|, α <4π, (14)

and as such it also appears in[3, Thm. 3.1], as a consequence of (6). The original Brezis–Merle inequality was obtained in[6]and it is essentially (14), but with a larger right-hand side. Similar inequalities without explicit right-hand side constants, but slightly more general integrands, were also obtained in[12], but either onW^2,1(Ω)or for functions ofW_,0^2,1(Ω)which are nonnegative or radial. The Brezis–Merle inequality quantifies the exponential integrability of functions inW_,0^2,1(Ω)andW^2,1(Ω), whenn=2; indeed it is well known that the functionuis inL_exp(Ω)if and only if

Ωe^λ|u|dx <∞, for someλ >0.

We observe that the discrepancy between the optimal ranges ofα’s in (13) and (14) is a phenomenon that is peculiar toL¹and the identities in (12). Indeed, the analogous sharp exponential inequality whenn >2

Ω

e^α

_|u(x)|

un/2 ⁿ

n−2

dxC, 0< αn(n−2)ⁿ⁻ⁿ²ω

2 n−2

n−1 (15)

was obtained by Adams[1]for the spaceW₀^2,n/2(Ω)=W_,0^2,n/2(Ω), but it can be easily extended to the larger space W^2,n/2(Ω)=W₀^1,p(Ω)∩W^2,p(Ω), with thesamesharp range ofα’s. The reason for that is that ifp >1 thenf⁺p

can be made arbitrarily close to fp within the class of functions with zero mean; the vanishing of the mean of uplays no role in (15), as opposed to the casen=2 in (13) and (14), where (12) causes a doubling of the largest exponential constant, going from general solutions of Dirichlet problems to compactly supported functions.

Whenn3 one instead obtains, as a result of (11), an estimate of type sup

u∈W_,0^2,1(Ω)

uq

u1 C

n, q,|Ω|

, 1q < n n−2

and a similar estimate forW^2,1(Ω), using (6). In Corollary 2 we will exhibit a specific constantC(n, q,|Ω|)which is sharp in the case ofW^2,1(Ω)andΩ a ball. Similar estimates without explicit constants, but slightly more general otherwise, were also obtained in[12], but again, only onW^2,1(Ω)or for functions ofW_,0^2,1(Ω)which are nonnegative or radial.

A question of interest that one can raise, in view of the embedding results in[12]and in the present paper, is the following:What is the smallest target space for the embeddings ofW_,0^2,1(Ω)?

A natural request in this sort of questions is that our admissible target spaces be the so-calledrearrangement in- variant spaces; those are Banach spaces(X, · X)of Lebesgue measurable functions onΩ with the property that uX= wX, wheneveruandw are equimeasurable. This problem has been fully investigated in the case of the classical Sobolev spaces embeddings. In particular, for the borderline embeddings ofW₀^{k,n/ k}(Ω)(n > k), the optimal r.i. target spaces turn out to be the so-called Hansson–Brezis–Wainger spaces[9,11,13,14,18]; such spaces are strictly contained in the exponential classes involved in the Adams–Moser–Trudinger inequalities[1]. See also[10], where op- timal embedding results are obtained for general Orlicz–Sobolev spaces, including those of Hansson–Brezis–Wainger as special cases.

The second main result of this paper is that the optimal target space for the embeddingW_,0^2,1(Ω) → X, whereX is an r.i. space overΩ, is the space of functions

Lexp,0(Ω)=

u∈Lexp(Ω): lim

t→0

u^∗∗(t) log¹_t =0

, whenn=2, and

L

n n−2,∞

0 (Ω)=

u∈Lⁿ−ⁿ2,∞(Ω): lim

t→0tⁿ⁻²ⁿ u^∗∗(t)=0

, whenn3, whereu^∗∗(t)=¹_tt

0u^∗(s) ds denotes the so-called maximal function ofu^∗. It is easy to see that the limit conditions in the above spaces can be unified as

tlim→0

u^∗(t)

N_|^∗_Ω|(t)=0 (16)

which is obviously a stronger condition than (11) from the point of view of “best target space”.

Whenn=2 the spaceL_exp,0(Ω)is a Banach subspace ofL_exp(Ω), endowed with the norm uLexp= sup

0<t|Ω|

u^∗∗(t)

1+log^|Ω_t^| (17)

and our optimal embedding result can be interpreted as the limiting case of the optimal borderline embeddings ob- tained by Hansson and Brezis–Wainger forW₀^{k,n/ k}(Ω),n > k.

Whenn3 the spaceL

n−2n ,∞

0 (Ω)is a Banach subspace ofL^n−n2,∞(Ω), endowed with the norm u_n−ⁿ₂,∞= sup

0<t|Ω|tⁿ⁻ⁿ²u^∗∗(t). (18)

The spaceL_exp,0(Ω)can also be characterized as the closure of the class of simple measurable functions onΩ, in the norm · Lexp, and also as the subspace of all order continuous elements ofL_exp(Ω)(i.e. thosef ∈L_exp(Ω)such

that if|f_n||f|and|f_n| ↓0 thenf_nLexp↓0). This is also true forL

n n−2,∞

0 (Ω), and in fact for any Marcinkiewicz space M_w(Ω), defined by the norm uMw =sup{u^∗∗(t)w(t)}, for a quasiconcave function w, and its subspace M_w⁰(Ω)= {u∈M_w(Ω): limt→0u^∗∗(t)w(t)=0}(see for example[16], and also[15]which contains a nice summary of the properties ofM_w⁰).

It is important to note that our optimal spacesLexp,0andL

n−2n ,∞

0 do not satisfy the so-called Fatou property, that is, they are not closed under a.e. limits of uniformly bounded sequences. For this reason the definition of r.i. space that we adopt here, given for example in[16], is the more general one, which does not require the Fatou property. It is an easy consequence of our result, however, that the optimal r.i. spaceswiththe Fatou property that containW_,0^2,1(Ω) areLexp(Ω), whenn=2, andL^nn−2,∞(Ω), whenn3 (see Theorem 2).

Our optimality results improve those obtained in Alberico and Cianchi[2], namely Theorem 1.1 in casek= +∞, n > p=2 and Theorem 1.2, (iii),k= +∞,n=p=2. In such theorems the authors prove in particular the optimality of the normsu_n−ⁿ₂,∞(n3) anduL_exp (n=2) in the inequality

uXCf1 (19)

among all r.i. spacesX satisfying the Fatou property, assuming that the inequality is valid for allf ∈L¹ and all solutionsuof a general class of boundary value problems, which includes the Dirichlet problem. Their proof is based on a duality argument and the fact that ifXis an r.i. space with the Fatou property then its second associate spaceX coincides withX. It is well known that ifXdoes not satisfy the Fatou property, thenXis a proper subspace ofX (see for example[7,16,15]for a summary of these and more facts on r.i. spaces, and references therein). In our result we assume only the minimal set of axioms for an r.i. space, and the validity of (19) whenf = −u, anducompactly supported inΩ, i.e. whenu∈W_,0^2,1(Ω).

Our proof is self-contained and borrows some ideas used in[11, Thm. 5], for the spacesW^{k,n/ k}. The key step is to prove that for a functionusatisfying (16) and with support inside a ball of volumeV one has

u^∗(t)(Tf )^∗(t), 0< tV

whereT is the Green potential for the ball, andf is a suitable positive radial function on the ball. This is a version of [11, Thm. 4]that is suited to our situation.

2. Sharp embedding inequalities forW^2,1(Ω)andW_,0^2,1(Ω)

IfΩis an open set ofRⁿandu:Ω→Ris Lebesgue measurable, the decreasing rearrangement ofuis the function u^∗(t)=inf

s0:x∈Ω: u(x)> st

, t >0

that is the function on[0,+∞)that is equimeasurable withuand also decreasing.

On a ballB_R=B(0, R)let NB_R(r)=

c_n(r²⁻ⁿ−R²⁻ⁿ) ifn3,

1

2πlog^R_r ifn=2, 0< rR with

cn= 1 (n−2)ωn−1

, ωn−1=2π^n/2 Γ (ⁿ₂).

IfBRis a ball of given volumeV and 0< tV, we let N_V^∗(t)=N_BR

nt ωn−1

1/n

= ₁

4πlog^V_t ifn=2,

c_nωn−1

n

ⁿ⁻²_n

(t⁻ⁿ⁻ⁿ²−V⁻ⁿ⁻ⁿ²) ifn3.

Note that ifG_BR(x, y)is the Green function for the ball of volumeV thenN_V^∗(t)is the decreasing rearrangement of G_BR(x,0).

Whenn3 we also set N_∞^∗(t)=c_n

ω_n−1 n

ⁿ⁻²_n

t⁻ⁿ⁻ⁿ², t >0. (20)

The-reduced spacesW^2,1(Ω)andW_,0^2,1(Ω)are defined in (1) and (2). Note that those definitions make sense for arbitrary open sets, not necessarily bounded. In particular whenΩ=Rⁿit is straightforward to check thatW^2,1(Rⁿ)= W_,0^2,1(Rⁿ).

Theorem 1.LetΩ⊆Rⁿ,n2, be open and bounded with volume|Ω|. Then:

(a) For allu∈W^2,1(Ω)

u^∗(t)N_|^∗_Ω|(t)u1, 0< t|Ω|. (21)

(b) For allu∈W_,0^2,1(Ω)andn2

u^∗(t)2^−2/nN_|^∗_Ω|(t)u1, 0< t|Ω| (22) and ifn3and eitheru0oruradial andΩa ball, then

u^∗(t)1

2N_|^∗_Ω|(t)u1, 0< t|Ω|. (23)

Whenn3both(22)and(23)hold forΩunbounded, with the convention in(20).

(c) The inequalities in(a)and(b)are sharp in the following sense:

sup

u∈X,0<t|Ω|

u^∗(t) N_|^∗_Ω|(t)u1=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

1 ifX=W^2,1(Ω), (24)

2^−2/n ifX=W_,0^2,1(Ω), (25)

1

2 ifX=W_,0^2,1(Ω)∩ {uradial}

orX=W_,0^2,1(Ω)∩ {u0}. (26)

Moreover, ifBis any ball, then for eacht∈(0,|B|]

sup

u∈X

u^∗(t) u1 =

⎧⎪

⎪⎨

⎪⎪

⎩

N_|^∗_B|(t) ifX=W^2,1(B), (27)

1

2N_|^∗_B|(t) ifX=W_,0^2,1(B)∩ {uradial},

orX=W_,0^2,1(B)∩ {u0} (28)

and also sup

u∈W_,0^2,1(Rⁿ)

u^∗(t) u1

=2^−2/nN_∞^∗(t). (29)

Remark.As we noted in the introduction, (21) appears in[3]and[12]and (23) appears in[12], in caseΩis smooth.

As an immediate consequence of Theorem 1 we obtain sharp norm embeddings for the spaces W^2,1(Ω)and W_,0^2,1(Ω). Recall that

L_exp(Ω)=

u:Ω→R, umeasurable andu^∗_Lexp<∞

(30) and

L^{n−2n ,∞}(Ω)=

u:Ω→R, umeasurable andu^∗n

n−2,∞<∞

, (31)

where the quasi-norms u^∗_Lexp andu^∗n

n−2,∞are defined as in (3). Note that in (30), (31) the normsuL_exp and u_n−ⁿ₂,∞defined in (17), (18) can be equivalently used in place of the corresponding quasi-norms.

Corollary 1.LetΩ⊆Rⁿ,n2, be open and bounded. Ifn=2thenW^2,1(Ω) →L_exp(Ω)and in particular u^∗_Lexp 1

4πu1, w∈W^2,1(Ω), (32)

u^∗Lexp 1

8πu1, w∈W_,0^2,1(Ω) (33)

and the constants_4π¹ and _8π¹ are sharp, i.e. they cannot be replaced by smaller constants.

Ifn3thenW^2,1(Ω) →L^{n−2n ,∞}(Ω)and in particular u^∗n

n−2,∞c_n ωn−1

n ⁿ⁻²

n u1, w∈W^2,1(Ω), (34)

u^∗n

n−2,∞2^−2/nc_n ωn−1

n ⁿ⁻²

n u1, w∈W_,0^2,1(Ω) (35)

and the constants are sharp.

Remark. Corollary 1 continues to hold if u^∗_Lexp and u^∗n

n−2,∞ are replaced by the larger quantities uL_exp, u_n−ⁿ₂,∞, and the constants in (32)–(35) are multiplied by ⁿ₂. The reason for this is that

N_V^∗∗(t)=1 t

t

0

N_V^∗(u) du= ₁

4π(1+log^V_t ) ifn=2,

cn(^ωn_n⁻¹)ⁿ⁻ⁿ²(ⁿ₂t⁻ⁿ⁻ⁿ² −V⁻ⁿ⁻ⁿ²) ifn3, so thatN_V^∗∗(t)∼ⁿ₂N_V^∗(t), ast→0.

Another immediate consequence of the estimates of Theorem 1 are the following sharp versions of the Brezis–

Merle and Maz’ya’s inequalities:

Corollary 2.LetΩ⊆Rⁿ,n2, be open and bounded. Ifn=2then

Ω

e^α|

u(x)|

u1 dx 4π

4π−α|Ω|, 0< α <4π, u∈W^2,1(Ω), (36)

Ω

e^α|

u(x)|

u1 dx 8π

8π−α|Ω|, 0< α <8π, u∈W_,0^2,1(Ω) (37) and the integrals are infinite ifα=4πin(36)andα=8πin(37). IfΩis a ball, the constants_4π^4π_−α,_8π^8π_−α are sharp.

Ifn3 then, for 1q <_n−ⁿ₂ uqc_n

ω_n−1 n

ⁿ⁻²

n Γ (_n−ⁿ₂−q)Γ (q+1) Γ (_n−ⁿ₂)

1/q

|Ω|^1q−n−n2u1, u∈W^2,1(Ω), (38)

uq2^−qn2 cn

ω_n−1 n

ⁿ⁻²

n Γ (_n−ⁿ₂−q)Γ (q+1) Γ (_n−ⁿ₂)

1/q

|Ω|^1q−n−2n u1, u∈W_,0^2,1(Ω) (39) and ifΩ is a ball, the constant is sharp in (38).

Proof of Theorem 1. The first step in the proof of (21) and (23) is Talenti’s comparison theorem, as in[3]and[12], and the following well-known formula for the solution of the Dirichlet problem−v=f on the ballB_R and with radial dataf∈L¹(B_R):

v

|x|

=NB_R

|x|

|y||x|

f (y) dy+

|x||y|R

NB_R

|y|

f (y) dy (40)

or, in polar coordinates, v(ρ)=ωn−1NB_R(ρ)

ρ

0

f (r)rⁿ⁻¹dr+ωn−1 R

ρ

NB_R(r)f (r)rⁿ⁻¹dr

= −ω_n−1

R

ρ

N_B

R(r) dr

r

0

f (ξ )ξⁿ⁻¹dξ. (41)

Note that if eitherf 0 orf decreasing with mean zero, thenv(ρ)given as in (41) is decreasing.

What we need here is the following version of Talenti’s result: letΩbe open and bounded and letf∈L¹(Ω)and let f(x)=f^∗(|B₁||x|ⁿ), the Schwarz symmetrization off, supported in the ballB_Rwith volume|Ω|; ifu, v∈W₀^1,1(Ω) are the unique solutions of−u=fand−v=f, thenu^∗(t)v^∗(t)fort >0. This result (including existence and uniqueness of the solutions) follows by a routine argument: (1) approximatef inL¹via a sequence off_n∈C₀^∞(Ω);

(2) solve the problems−u_n=f_n,−v_n=f_n; (3) use the uniform gradient estimate∇u_n1∇v_n1Cf1

(the left inequality for example is in [19, p. 715]); (4) show that {u_n} is a Cauchy sequence convergent tou, the solution of−u=f; (5) apply Talenti’s classical result to theun, and pass to the limit.

To prove (21) we then apply the above version of Talenti’s theorem to a function∈W^2,1(Ω), and conclude that u^∗(t)v^∗(t)for t >0, where v is the solution of −v=(u),v=0 on∂B_R. Next, note that the solution of

−v=f (v=0 on∂B_R) withf radial given in (40) satisfies v

|x|N_BR

|x| f1

which instantly gives (21).

A small modification of the above argument yields (23) in the caseu∈W_,0^2,1(Ω)with either u0 oruradial.

Indeed, assuming WLOG thatu∈C₀^∞(Ω), then

Ωu=0, so lettingf = −u, andf⁺,f⁻be the positive and negative parts off, we have

Ωf⁺=

Ωf⁻=¹₂f1. Ifuis radial then (40) yields

−NB_R

|x|

B_R

f⁻(y) dyv

|x| NB_R

|x|

B_R

f⁺(y) dy or

v

|x|1 2N_BR

|x| f1

from which (23) follows. Ifu0 then lettingwbe the solution of−w=f⁺ onΩ, withw∈W₀^1,1(Ω)we have 0uw, by the maximum principle, and the result follows from part (a) applied tow.

To prove (22) we argue as follows. First, note that it is enough to prove the result foru∈C^∞₀ (Ω). For such given uand for each0 consider the open subsets ofΩ

Ω=

x∈Ω: u(x) >

, Ω=

x∈Ω: −u(x) >

and the functions

u:=(u−)|Ω, u=(−u−)|Ω.

Sard’s theorem combined with the implicit function theorem guarantee that for a.e. >0 both∂Ωand∂Ω are smooth C^∞ (n−1)-dimensional manifolds; therefore, for each such bothu and u are C^∞ in their domains, continuous up to the boundaries, and with zero boundary values, and if f = −u they clearly solve the Dirichlet problems−u=f and−u= −f in their domains. Let noww_e,wbe the solutions to the Dirichlet problems

−w=f⁺ onΩ, w=0 on∂Ω,

−w=f⁻ onΩ, w=0 on∂Ω.

Then we have 0uwand 0uw, and alsow∈W^1,2(Ω),w∈W^1,2(Ω). We can then apply part (a) to deduce

(u)^∗(t)(w)^∗(t)N_|^∗_Ω

|(t)

Ω

f⁺dx,

for 0< t|Ω|and hence for 0< t|Ω₀|. All the quantities involved above are monotone decreasing w.r.t.hence we deduce

(u₀)^∗(t)N_|^∗_Ω

0|(t)

Ω0

f⁺=1 2N_|^∗_Ω

0|(t)u1, 0< t|Ω₀|. (42) Likewise, arguing withu,w, we obtain

u_0∗ (t)1

2N_|^∗_Ω

0|(t)u1, 0< tΩ₀. (43)

Let nowλV(s)be the distribution function ofN_V^∗, i.e.

λ_V(s)=t >0:N_V^∗(t) > s=

V e^{−4π s} ifn=2, (αns+V⁻ⁿ⁻ⁿ²)⁻ⁿ⁻ⁿ² ifn3 whereαn=(n−2)nⁿ⁻ⁿ²ω^2/nn . With this notation we have, fors >0,

x∈Ω: u(x)> s=x∈Ω0: u0(x) > s+x∈Ω₀: u₀(x) > s λ_|Ω₀|

2s u1

+λ_|Ω

0|

2s u1

. (44)

Now note that|Ω₀| + |Ω₀| = |Ω|and that λ_|Ω₀|

2s u1

+λ_|Ω

0|

2s u1

|Ω|e^{−8π s/u1} ifn=2, (2^{2/n α}_u^ns

1+ |Ω|⁻ⁿ⁻ⁿ²)⁻ⁿ⁻ⁿ² ifn3, (45)

since forn=2 there actually is equality, whereas forn3 the right-hand side of (44) is maximized precisely when

|Ω₀| = |Ω₀| =¹₂|Ω|. Inequalities (44) and (45) imply (22).

Now let us prove the sharpness statements. Introduce the radially decreasing functions F_δ^R= χ_Bδ

|B_δ|, 0< δ < R, F_δ,^R = χB_δ

2|Bδ|− χA_,R

2|A,R|, 0< δ < R−2 < R where

Bδ=

x: |x|δ

, A,R=

x: R−2 <|x|< R− .

Applying formula (40) we obtain that the solutionU_δ^Rof the Dirichlet problem −U_δ^R=F_δ^R onB_R,

U_δ^R=0 on∂B_R is given by

U_δ^R(x):=

_|x|n

δⁿ N_BR(|x|)+_|B¹_δ|

|x|<|y|<δN_BR(|y|) dy if|x|< δ, N_BR(|x|) ifδ|x|R, which is nonnegative, radial and decreasing, so that

U_δ^R_∗

(t)=N_|^∗_B

R|(t), |B_δ|t|B_R|,

and this takes care of (27) immediately, sinceU_δ^R∈W^2,1(BR).

If Ω is an arbitrary open and bounded set, then we can assume that 0∈Ω, and findR so that BR⊆Ω. The function U_δ^R (extended to be 0 outside B_R) is not inW^2,1(Ω), however we can argue that sinceF_δ^R0 then the solutionU_δ∈W^2,1(Ω)of−U_δ=F_δ^Ris nonnegative onΩand satisfiesU_δ^RU_δonB_R, by the maximum principle;

hence(U_δ)^∗(t)(U_δ^R)^∗(t)=N_|^∗_B

R|(t), for|B_δ|t|B_R|. It’s then clear that takingδ_tso that|B_δt| =tgives (U_δt)^∗(t)

N_|^∗_Ω|(t) N_|^∗_B

R|(t)

N_|^∗_Ω|(t) →1, t→0, thereby proving (24).

Likewise, the solutionU_δ,^R to −U_δ,^R =F_δ,^R onB_R,

U_δ,e^R =0 on∂B_R

can be computed explicitly, however all we need is thatU_δ,^R is nonnegative, radial, decreasing on(0,|B_R|], and U_δ,^R(x)=

₁

2N_BR(|x|)−_2|A¹_,R|

A,RN_BR(|y|) dy ifδ|x|R−2,

0 ifR−|x|R (46)

all of which can be readily checked. We then haveU_δ,^R ∈W_,0^2,1(B(0, R)), and the above identity leads to (28), since

lim→0

1 2|A_,R|

A_,R

NB_R

|y| dy=0.

For an arbitrary open and bounded Ω, we can prove (26) like before, assuming 0∈Ω, B(0, R)⊆Ω, this time observing that U_δ,^R ∈W_,0^2,1(B(0, R))⊆W_,0^2,1(Ω). It remains to settle (25) and (29) for n3. We consider the functions

V_δ,λ^R (x)=U_δ,R/4^R (x)−U_δ,R/4^R (x−xλ), xλ:=(λ,0,0, . . . ,0), with

δ <min 1

2,1 2R

, δ <1 2λ <1

2R, (47)

so that

−V_δ,λ^R = 1 2|Bδ|

χ_Bδ−χ_xλ+Bδ

−h^R_λ, whereBδandBδ+xλare disjoint and where

h^R_λ = 1

|A_R/4,R|(χA_R/4,R−χx_λ+A_R/4,R)

which converges to 0 pointwise and inL¹, asλ→0 for fixedR, and asR→ +∞for fixedλ; moreover,|h^R_δ|CR⁻ⁿ and

Rⁿ

h^R_λCλ

R. (48)

Note thatV_δ,λ^R ∈W_,0^2,1(B(0, R+λ)).