Brunn-Minkowski inequalities for two functionalsinvolving the p-Laplace operator

Andrea Colesanti ∗ Paola Cuoghi† Paolo Salani‡

Abstract

In the family of n-dimensional convex bodies, we prove a Brunn-Minkowski type in-equality for the first eigenvalue of the p-Laplace operator, or Poincare constant, andfor a functional extending the notion of torsional rigidity. In the latter case we alsocharacterize equality conditions.

Keywords and phrases: Brunn-Minkowski inequality, p-Laplace operator, torsional rigid-ity, first eigenvalue.

AMS 2000 Subject Calssification: 35J60, 39B62, 47J05, 52A40, 52C25.

1 Introduction

This paper represents a continuation of our previous work, contained in [9], [8], [19] and [7],on Brunn-Minkowski type inequalities for functionals of the Calculus of Variations.

We start by recalling the Brunn-Minkowski inequality in its classical formulation. LetK0 and K1 be compact convex sets in Rn with non-empty interior, i.e. convex bodies, andfix t ∈ [0, 1]; then consider the convex linear combination of these sets:

Kt = (1− t)K0 + tK1 = (1− t)x + ty |x ∈ K0 , y ∈ K1 ,

which is still a convex body.The Brunn-Minkowski inequality claims that

V (Kt)1/n ≥ (1− t)V (K0)

1/n + tV (K1)1/n , (1)

where V denotes the n-dimensional volume (i.e. the Lebesgue measure). Moreover, equalityholds in (1) if and only if K0 and K1 are homothetic, i.e. they are equal up to translationand dilatation.

∗Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze, Italy; cole-sant@math.unifi.it

†Dipartimento di Matematica “G. Vitali”, via Campi 213/b, 41100 Modena, Italy, pcuoghi@unimo.it‡Dipartimento di Matematica “U. Dini”, viale Morgagni 67/A, 50134 Firenze, Italy; salani@math.unifi.it

1

As bibliographical references about this inequality, we suggest the book by Schneider[20] (cfr Chapter 6) and the survey paper by Gardner [11]. The Brunn-Minkowski inequalityhas a fundamental role in the theory of convex bodies, moreover it is strongly connected toother inequalities like the isoperimetric inequality and the Sobolev inequality. Notice thatthe validity of (1) goes far beyond the family of convex bodies: namely it can be extendedto the class of measurable sets (see [11] for more details).

LetKn denote the class of convex bodies in Rn; Kn is endowed with a scalar multiplicationfor positive numbers:

s K = s x |x ∈ K , K ∈ Kn , s > 0 ,

and with the Minkowski addition:

K0 + K1 = x + y |x ∈ K0 , y ∈ K1 , K0, K1 ∈ Kn .

The Brunn-Minkowski inequality is equivalent to the fact that the n-dimensional volumeraised to the power 1/n is concave in Kn; note that V is positively homogeneous and itsorder of homogeneity is precisely n:

V (s K) = sn V (K) , ∀s > 0 .

These considerations lead us to the following:

Definition 1.1 Let F : Kn → R+ be a functional, invariant under rigid motions of Rn. IfF is positively homogeneous of some order α 6= 0 and F 1/α is concave in Kn, then we saythat F satisfies a Brunn-Minkowski type inequality.

In convex geometry there are many examples of functionals satisfying a Brunn-Minkowskiinequality: the (n−1)-dimensional measure of the boundary, the quermassintegrals, etc. (see[20] and [11]). On the other hand, inequalities of Brunn-Minkowski type have been provedfor functionals coming from a quite different area: the Calculus of Variations.

The first example in this sense is due to Brascamp and Lieb who proved that the firsteigenvalue of the Laplace operator satisfies a Brunn-Minkowski inequality (cfr [5]). Subse-quently, Borell proved the same result for the Newton capacity, the logarithmic capacity (indimension n = 2) and the torsional rigidity (cfr [2], [3] and [4] respectively). In [6] Caffarelli,Jerison and Lieb established equality conditions for the Newton capacity. Let us notice thatfor the first eigenvalue of the Laplacian and the torsional rigidity, the Brunn-Minkowskiinequality extends to the class of connected domains (with sufficiently smooth boundary)while so far this has not been proved for the capacity.

These results have been generalized, improved and developed in various directions: in [9]it is proved that the p-capacity, p ∈ (1, n), satisfies a Brunn-Minkowski inequality (includingequality conditions); the same result is proved in [8] for a n-dimensional version of thelogarithmic capacity and in [19] for the eigenvalue of the Monge-Ampere operator. Moreoverin [7] equality conditions are established for the first eigenvalue of the Laplacian (only in thecase of convex bodies) and the torsional rigidity. Notice that in all known cases, equalityconditions are the same as in the Brunn-Minkowski inequality for the volume, i.e. equalityholds if and only if the involved sets are (convex and) homothetic.

2

In this paper we make a further progress, proving a Brunn-Minkowski inequality for thefirst eigenvalue of the p-Laplace operator, or Poincare constant, p > 1, and a for functionalwhich extends the notion of torsional rigidity, that we will call p-torsional rigidity.

Let K be a convex body and assume that its boundary is of class C2, let Ω be the interiorof K and fix p > 1. The Poincare constant λ(K) of K is defined as follows

λ(K) = inf

∫Ω|∇v|pdx∫

Ω|v|pdx

: v ∈ W 1,p0 (Ω) ,

∫Ω

|v|pdx > 0

. (2)

Sakaguchi [18] proved that the infimum is attained for functions belonging to W 1,p0 (Ω), which

are weak solutions of ∆pu = −λ(K)|u|p−2u in Ωu = 0 on ∂Ω ,

(3)

where∆pu = div(|∇u|p−2∇u)

is the p-Laplace operator. For this reason λ is also termed the first eigenvalue of the p-Laplace operator; in particular, for p = 2 we have the first eigenvalue of the ordinary Laplaceoperator.

An immediate consequence of its definition is that λ is positively homogeneous of order−p.

We prove the following

Theorem 1.1 Let K0 and K1 be convex bodies in Rn with boundary of class C2 and letp > 1. For t ∈ [0, 1], let Kt = (1− t)K0 + tK1. Then, the following inequality holds

λ(Kt)− 1

p ≥ (1− t)λ(K0)− 1

p + tλ(K1)− 1

p . (4)

The proof of this result follows the idea used in [7] for the case p = 2. Our method allows toobtain a partial result regarding equality conditions: if there is equality in (4) and at leastone of the following conditions holds:

1. n = 2,

or

2. K0 and K1 have positive Gauss curvature at each point of their boundary,

then K0 and K1 are homothetic.

Now we describe the result for the p-torsional rigidity. Let K be a convex body in Rn

and let Ω represents its interior. For p > 1, we define the p-torsional rigidity of K, τ(K),through the following formula:

1

τ(K)= inf

∫Ω|∇w(x)|pdx[∫

Ω|w(x)|dx

]p : w ∈ W 1,p0 (Ω) ,

∫Ω

|w(x)|dx > 0

. (5)

3

For n = p = 2 we obtain the usual notion of torsional rigidity, see for instance the book byPolya and Szego [16].

The p-torsional rigidity is homogeneous of degree p + n(p− 1).We have the following:

Theorem 1.2 Let K0 and K1 be convex bodies in Rn, and p > 1; for t ∈ [0, 1], let Kt =(1− t)K0 + tK1. Then the following inequality holds

τ(Kt)1

p+n(p−1) ≥ (1− t)τ(K0)1

p+n(p−1) + tτ(K1)1

p+n(p−1) . (6)

Equality occurs if and only if K1 is homothetic to K0.

Note that in this case we do not need any additional assumption on the regularity of ∂K.This result holds also under different assumptions on the geometry of the involved sets; seethe remark at the end of §4.

As in the case p = 2 (see [7]), the proof of Theorems 1.2 is based on a technique employedto prove quasi-concavity (more precisely, power-concavity) of solutions to elliptic equations,due to Korevaar and Kennington (cfr [14] and [13]). In particular, we extend the Korevaarconcavity maximum principle to the case of three functions, instead of one. One technicaldifficulty is the degeneracy of the operator ∆p, for which we use the approximation argumentpresented by Sakaguchi in [18].

2 Preliminaries

Throughout the paper, K (possibly with subscripts) denotes a convex body in Rn, that isa convex, compact set with non-empty interior, and Ω (possibly with subscripts) representsthe interior of K. We say that a convex body is of class C2,+ if its boundary ∂K is of classC2 and the Gauss curvature at every point of ∂K is strictly positive.

Let u : K → R be a twice differentiable function; we denote by ∇u = ( ∂u∂x1

, ..., ∂u∂xn

) the

gradient of u and by D2u =(

∂2u∂xi∂xj

)ij

its Hessian matrix. We say that u is of class C2,−(Ω),

for some open set Ω, if u ∈ C2(Ω) and D2u(x) < 0 for every x ∈ Ω.Let us consider two open convex sets Ω0 and Ω1 in Rn and t ∈ [0, 1]. For i = 0, 1, let

ui : Ωi → R be a concave function.The sup-convolution u of u0 and u1 is defined in Ωt = (1− t)Ω0 + tΩ1 as follows

u(z) = sup (1− t)u0(x) + tu1(y) : x ∈ Ω0, y ∈ Ω1, z = (1− t)x + ty .

Remark 2.1 Observe that −u coincides with the infimal convolution of −u0 and −u1, as itis defined in Section 5 of [17]. In particular, as the infimal convolution of convex functionsis convex, u turns out to be concave.

We prove two auxiliary results that will be used in the proof of Theorem 1.1.

4

Lemma 2.1 For i = 0, 1, let Ωi be an open, bounded, convex set and ui ∈ C1(Ωi) bea strictly concave function such that limx→∂Ωi

ui(x) = −∞. Then, for t ∈ [0, 1] (with thenotation introduced above), u ∈ C1(Ωt) and it is strictly concave; moreover, for every z ∈ Ωt,there exists a unique couple of points (x, y) ∈ Ω0 × Ω1 such that

z = (1− t)x + ty , (7)

u(z) = (1− t)u0(x) + tu1(y) , (8)

∇u(z) = ∇u0(x) = ∇u1(y) . (9)

If in addition z ∈ Ωt is such that u0 and u1 are twice differentiable at the corresponding pointx and y respectively, and D2u0(x) , D2u1(y) < 0, then u is twice differentiable at z and

D2u(z) =[(1− t)

(D2u0(x)

)−1+ t

(D2u1(y)

)−1]−1

. (10)

Proof. We fix z ∈ Ωt. As, for i = 0, 1, ui goes to −∞ on the boundary of Ωi, the supremumin the definition of u is in fact a maximum. Now, assume that there exist two distinct couples(xa, ya), (xb, yb) ∈ Ω0 × Ω1 such that (7) and (8) hold and define a third couple

(xc, yc) =1

2[(xa, ya) + (xb, yb)] ∈ Ω0 × Ω1 .

¿From the definition of u and by the strict concavity of u0 and u1 it follows

u(z) ≥ (1− t)u0(xc) + tu1(yc) > (1− t)1

2[u0(xa) + u0(xb)] + t

1

2[u1(ya) + u1(yb)] = u(z)

i.e. a contradiction. The function u goes to −∞ on the boundary of Ωt; indeed, let zk, k ∈ N,be a sequence of points in Ωt tending to ∂Ωt and let xk and yk be determined correspondinglyin Ω0 and Ω1 as above. Up to subsequences, we may assume that zk → z ∈ Kt, xk → x ∈ K0

and yk → y ∈ K1. In particular, z ∈ ∂Kt so that, by (7), x ∈ ∂K0 and y ∈ ∂K1 and then(8) implies that u(zk) tends to −∞ as k → +∞. A further consequence is that

(∂(−u))(Ωt) = Rn , (11)

(here ∂ denotes the usual sub-differential of convex functions); similarly (∇ui)(Ωi) = Rn,for i = 0, 1. ¿From Theorem 16.4 in [17] (see also Remark 2.1 in this section) we have that

(−u)∗ = (1− t)(−u0)∗ + t(−u1)

∗ , (12)

where the symbol ∗ means the usual conjugation of convex functions (cfr Section 12 in [17]).Following the notation of [17], we have that −u0 and −u1 are essentially smooth and strictlyconvex (by the assumptions of the lemma) so that by [17, Theorem 26.3] the same propertiesalso hold for their conjugate functions. Now, by (12) and [17, Theorem 26.3], −u is strictlyconvex and essentially smooth and from [17, Corollary 25.5.1], u ∈ C1(Ωt). Let z ∈ Ωt

and x and y be such that (7) and (8) hold; a straightforward consequence of the Lagrangemultipliers Theorem is that

∇u0(x) = ∇u1(y) .

5

Let X = ∇u0(x) = ∇u1(y); by [17, Theorem 26.5] and (12), we have

(∇u)−1(X) = (1− t)(∇u0)−1(X) + t(∇u1)

−1(X) = (1− t)x + ty = z . (13)

Since we have proved that u is strictly concave, its gradient map is invertible and the lastformula implies (9).

Now let z ∈ Ωt be such that u0 and u1 are twice differentiable at the corresponding pointsx and y respectively, and assume that D2u0(x) , D2u1(y) < 0; let X = ∇u0(x) = ∇u1(y) =∇u(z). Then the mappings (∇u0)

−1 and (∇u1)−1 are differentiable at X and

D(∇u0)−1(X) = [D2u0(x)]−1 < 0 , D(∇u1)

−1(X) = [D2u1(y)]−1 < 0 .

Now using (13) we have that (∇u)−1 is differentiable at X and the determinant of its Jacobianmatrix at X does not vanish, so that u is twice differentiable at z. Formula (10) is again asimple consequence of the first equality in (13).

Lemma 2.2 Let Ω0 and Ω1 be two open bounded convex sets and let ui : Ωi → R, ui ∈C1(Ωi), for i = 0, 1, be a strictly concave function. Fix t ∈ [0, 1], for a point z ∈ Ωt let(x, y) ∈ Ω0×Ω1 be the unique couple of points determined by Lemma 2.1. If ∇u(z) 6= 0, andu0 and u1 are twice differentiable at x and y respectively, with D2u0(x) , D2u1(y) < 0, then

∆pu(z) ≥ (1− t)∆pu0(x) + t∆pu1(y) . (14)

Proof. For a generic (smooth) function u and a point x such that ∇u(x) 6= 0, we may write

∆pu(x) = |∇u(x)|p−2

(∆u(x) + (p− 2)

∂2u(x)

∂ν2

), (15)

where ν = ∇u(x)/|∇u(x)| and

∂2u(x)

∂ν2=

⟨D2u(x) ν, ν

⟩.

We choose a coordinate frame such that

en =∇u0(x)

|∇u0(x)|=

∇u1(y)

|∇u1(y)|=

∇u(z)

|∇u(z)|,

where e1, e2, . . . , en represent the standard basis of Rn. Let α = |∇u0(x)| = |∇u1(y)| =|∇u(z)|, from (15) we have

∆pu(z) = αp−2(trace D2u(z) + (p− 2)

⟨D2u(z)en, en

⟩),

∆pu0(x) = αp−2(trace D2u0(x) + (p− 2)

⟨D2u0(x)en, en

⟩),

∆pu1(y) = αp−2(trace D2u1(y) + (p− 2)

⟨D2u1(y)en, en

⟩).

So, to establish (14) it suffices to prove:i) trace D2u(z) ≥ (1− t)trace D2u0(x) + t trace D2u1(y),ii) (D2u(z)en, en) ≥ (1− t) (D2u0(x)en, en) + t (D2u1(y)en, en).

These formulas easily follows from (10) and from inequality (37) in [1] (notice that here theinequalities are reversed with respect to those in [1] since we are considering negative definitematrices).

6

3 Proof of Theorem 1.1

Let K be a convex body in Rn, with boundary of class C2 and denote by u a weak solution ofproblem (3), the existence of such solution is proved in [18]. We start by some considerationsregarding u. Firstly, any multiple of u is also a solution of (3); on the other hand, all thesolutions are proportional to each other, i.e. the family of the solutions is a one-dimensionalvector space. Concerning the regularity of u, we have u ∈ C1,α(K) for some constantα ∈ (0, 1) (cfr Theorem A.1 in [18]). Moreover, Theorem 1.1 in [18] claims that if u is anypositive solution of (3), then the function v = log u is concave.

Remark 3.1 Consider the set

Ω = x ∈ Ω| ∇u = 0 ,

where u is a non-trivial solution of (3) in Ω; as all the solutions are proportional to each

other, this set depends only on Ω. Ω is convex because it is a level set of a log-concavefunction; more precisely it is the set where u attains its maximum value in K.

Remark 3.2 If u is any non-trivial solution of (3), then the p-Laplace operator (applied

to u) is uniformly elliptic on compact subsets of Ω\Ω. By standard regularity results for

solution of elliptic equations, we have that u ∈ C2(Ω\Ω); hence the function v = log u solves∆pv = −[λ + (p− 1)|∇v|p ] in Ω\Ωv → −∞ on ∂Ω.

(16)

Proof of Theorem 1.1. In order to prove (4), we first establish the following inequality

λt ≤ (1− t)λ0 + tλ1, (17)

where λi is the Poincare constant of Ki, for i = 0, 1, t. Clearly, from (17), we have that

λt ≤ maxλ0, λ1. (18)

This fact, together with the homogeneity of λ, proves (4) by the following standard argument:for arbitrary K0 and K1 and t ∈ [0, 1], let

K ′0 = [λ(K0)]

1/pK0 , K ′1 = [λ(K1)]

1/pK1 , t′ =t[λ(K1)]

−1/p

(1− t)[λ(K0)]−1/p + t[λ(K1)]−1/p

and apply (18) to K ′0, K ′

1 and t′; (4) follows.

In the sequel, we denote by ui a positive solution of (3) in Ωi, for i = 0, 1, t. We knowthat the function

vi = log ui

is concave in Ωi, it belongs to C2(Ωi \ Ωi) and it is a solution of (16) in Ωi \ Ωi. Let us denoteby v the sup-convolution of v0 and v1, as defined in §2:

v(z) = sup (1− t)v0(x) + tv1(y) : x ∈ Ω0, y ∈ Ω1, z = (1− t)x + ty .

7

¿From [17, Corollaries 26.3.2 and 25.5.1], it follows that v ∈ C1(Ωt). We construct a sequenceof new functions approximating vi, i = 0, 1, and which satisfies the assumptions of Lemma2.2. For ε > 0 we define

vi,ε(x) = vi(x)− ε|x|2

2, x ∈ Ωi .

The function vi,ε is strictly concave in Ωi and

vi,ε ∈ C2,−(Ωi\Ωi), for i = 0, 1. (19)

We consider the sup-convolution vε of v0,ε, v1,ε, that is

vε(z) = sup (1− t)v0,ε(x) + tv1,ε(y) : x ∈ Ω0, y ∈ Ω1, z = (1− t)x + ty . (20)

Clearly vi,ε converges uniformly to vi in Ωi, for i = 0, 1, but we can also see that vε convergesuniformly to v in Ωt. Indeed, from the definition of vε and vi,ε, we have that

vε ≤ v ; (21)

on the other hand

vε ≥ v(z)− ε

2sup

(1− t)|x|2 + t|y|2 : x ∈ Ω0, y ∈ Ω1, z = (1− t)x + ty

= v(z)− Cε ,

where C > 0 is a constant independent of ε. The last equality together with (21) gives us

|vε(z)− v(z)| ≤ Cε, ∀z ∈ Ωt ,

that is the uniform convergence. Actually, we can say more: from [17, Theorem 25.7] weconclude that

∇vi,ε converges uniformly to ∇vi on every compact subset of Ωi , i = 0, 1 , (22)

∇vε converges uniformly to ∇v on every compact subset of Ωt . (23)

The next step is to express the p-Laplacian of vi,ε in terms of the p-Laplacian of vi. For

i = 0, 1 and x ∈ Ωi \ Ωi, we put

νi,ε(x) =∇vi,ε(x)

|∇vi,ε(x)|,

so, by formula (15), we obtain

∆pvi,ε(x) = |∇vi,ε(x)|p−2(∆vi,ε(x) + (p− 2)

⟨D2vi,ε(x)νi,ε, νi,ε

⟩)= |∇vi(x)− εx|p−2

(∆vi(x)− nε + (p− 2)

⟨D2vi(x)νi,ε, νi,ε

⟩− ε(p− 2) 〈Inνi,ε, νi,ε〉

)= |∇vi(x)− εx|p−2

(∆vi(x) + (p− 2)

⟨D2vi(x)νi,ε, νi,ε

⟩− ε(n + p− 2)

);

or

8

∆pvi,ε(x) = ∆pvi(x) +(|∇vi(x)− εx|p−2 − |∇vi(x)|p−2

)∆vi(x) +

+(p− 2)[|∇vi(x)− εx|p−2

⟨D2vi(x) νi,ε, νi,ε

⟩− |∇vi(x)|p−2

⟨D2vi(x)νi, νi

⟩]+

−ε(n + p− 2)|∇vi(x)− εx|p−2. (24)

LetΩt = x ∈ Ωt : ∇v = 0 ;

from [17, Theorems 26.5 and 23.8] and (12) we have that

Ωt = ∂(−v∗(0)) = ∂ ((1− t)(−v∗0(0)) + t(−v∗1(0)))

= (1− t)∂(−v∗0(0)) + t∂(−v∗1(0)) = (1− t)Ω0 + tΩ1,

(25)

where, in the last equality, we used the fact that Ωi = ∇ui = 0 = ∇vi = 0. ¿FromLemma 2.1 and (19) it follows

vε ∈ C2,−(Ωt\Ωt) .

Moreover, from the same lemma, for a fixed z ∈ Ωt\Ωt, there exists a unique (xε, yε) ∈Ω0 × Ω1, depending on z, such that z = (1− t)xε + tyε and

∇vε(z) = ∇v0,ε(xε) = ∇v1,ε(yε) . (26)

As we know that ∇v(z) 6= 0, from (22), (23) and the last equality, we can deduce that thereexists ε1 > 0 such that, for 0 < ε < ε1,

(xε, yε) ∈((Ω0\Ω0)× (Ω1\Ω1)

).

Now we apply Lemma 2.2, so

∆pvε(z) ≥ (1− t)∆pv0,ε(xε) + t∆pv1,ε(yε) .

With the help of (24) we get

∆pvε(z) ≥ (1− t)∆pv0(xε) + t∆pv1(yε) + (1− t)∆v0(xε)×

×(|∇v0(xε)− εxε|p−2 − |∇v0(xε)|p−2

)+ (1− t)(p− 2)×

×[|∇v0(xε)− εxε|p−2 ⟨

D2v0(xε)ν0,ε, ν0,ε

⟩− |∇v0(xε)|p−2

⟨D2v0(xε)ν0, ν0

⟩]+

+t∆v1(yε)(|∇v1(yε)− εyε|p−2 − |∇v1(yε)|p−2

)+ t(p− 2)×

×[|∇v1(yε)− εyε|p−2 ⟨

D2v1(yε)ν1,ε, ν1,ε

⟩− |∇v1(yε)|p−2

⟨D2v1(yε)ν1, ν1

⟩]+

−ε(n + p− 2)[(1− t)

(|∇v0(xε)− εxε|p−2 + t

(|∇v1(yε)− εyε|p−2

))].

9

Set

Fε(z) = (1− t)∆v0(xε)(|∇v0(xε)− εxε|p−2 − |∇v0(xε)|p−2

)+ (1− t)(p− 2)×

×[|∇v0(xε)− εxε|p−2 ⟨

D2v0(xε)ν0,ε, ν0,ε

⟩− |∇v0(xε)|p−2

⟨D2v0(xε)ν0, ν0

⟩]+

+t∆v1(yε)(|∇v1(yε)− εyε|p−2 − |∇v1(yε)|p−2

)+ t(p− 2)×

×[|∇v1(yε)− εyε|p−2 ⟨

D2v1(yε)ν1,ε, ν1,ε

⟩− |∇v1(yε)|p−2

⟨D2v1(yε)ν1, ν1

⟩]+

−ε(n + p− 2)[(1− t)

(|∇v0(xε)− εxε|p−2 + t

(|∇v1(yε)− εyε|p−2

))],

so that∆pvε(z) ≥ (1− t)∆pv0(xε) + t∆pv1(yε) + Fε(z) ,

which, by (16), implies

∆pvε(z) ≥ − [(1− t)λ0 + tλ1 + (p− 1)((1− t)|∇v0(xε)|p + t|∇v1(yε)|p)] + Fε(z) . (27)

¿From the definition of v0,ε, v1,ε and (26) we get

∆pvε(z) ≥ − [(1− t)λ0 + tλ1 + (p− 1)((1− t)|∇vε(z) + εxε|p + t|∇vε(z) + εyε|p)] + Fε(z)

= − [(1− t)λ0 + tλ1 + (p− 1)|∇vε(z)|p ] +

+(p− 1) [ |∇vε(z)|p − (1− t)|∇vε(z) + εxε|p − t|∇vε(z) + εyε|p ] + Fε(z) .

We set

Fε(z) = Fε(z) + (p− 1) [ |∇vε(z)|p − (1− t)|∇vε(z) + εxε|p − t|∇vε(z) + εyε|p ] ,

so that we can write

∆pvε(z) ≥ − [(1− t)λ0 + tλ1 + (p− 1)|∇vε(z)|p ] + Fε(z) . (28)

ClearlyFε → 0 point-wise in Ωt\Ωt .

Moreover, if C is a compact subset of Ωt\Ωt, then there exist ε = ε(C) and two compact

subsets C0 and C1 of Ω0\Ω0 and Ω1\Ω1 respectively, such that, for every z ∈ C and for0 < ε < ε,

(xε, yε) ∈ C0 × C1 . (29)

This follows from (26) and the definition of Ωi. So, as vi ∈ C2(Ci), for i = 0, 1, we have

that |Fε| is uniformly bounded on C with respect to ε < ε; consequently the sequence Fε is

uniformly bounded on compact subsets of Ωt\Ωt.We can now define the functions

uε(z) = evε(z) , z ∈ Ωt , ε > 0 .

10

Note that as vε → −∞ on the boundary, uε ∈ C(Kt) and it vanishes on ∂Ωt. Furthermore,from Lemma 2.1, for every z ∈ Ωt ,

|∇uε(z)| = |∇vε(z)|evε(z) = |∇v0,ε(xε)|1−t|∇v1,ε(yε)|te(1−t)v0,ε(xε)+tv1,ε(yε)

=[|∇v0,ε(xε)|ev0,ε(xε)

]1−t [|∇v1,ε(yε)|ev1,ε(yε)]t

= |∇(ev0,ε(xε)

)|1−t|∇

(ev1,ε(yε)

)|t .

¿From the last equality, the definition of v0,ε and v1,ε and the regularity of v0 and v1 (cfr[18]), we obtain that |∇uε| is bounded in Kt, and, in particular, uε ∈ W 1,p

0 (Ωt).¿From (28) we get

∆puε(z) ≥ − [(1− t)λ0 + tλ1] uε(z)p−1 + Fε(z)uε(z)p−1 , z ∈ Ωt \ Ωt . (30)

Now, let us consider a sequence of compact sets Tj = Aj\Bj, where Aj and Bj are open

convex sets so that Aj ⊂⊂ Ωt, Bj ⊃ Ωt and Bj ⊂ Aj. Moreover, assume that Aj → Ωt and

Bj → Ωt in the Hausdorff metric, as j → +∞. For every j ∈ N, we multiply each side ofinequality (30) by uε(z) and we integrate it over Tj:∫

Tj

uε(z)∆puε(z)dz ≥ − [(1− t)λ0 + tλ1]

∫Tj

uε(z)pdz +

∫Tj

Fε(z)uε(z)pdz .

Integrating by parts we get∫Tj

|∇uε(z)|pdz −∫

∂Tj

uε(z)|∇uε(z)|p−2∂uε(z)

∂νdz ≤

(31)

≤ [(1− t)λ0 + tλ1]

∫Tj

uε(z)pdz −∫

Tj

Fε(z)uε(z)pdz .

Note that, as vε converges to v uniformly, then uε converges to u = ev uniformly in Kt.Moreover, from (22) and (23), ∇uε converges to ∇u uniformly on Tj. Passing to the limit

for ε → 0 in (31), using the uniform convergence of uε and ∇uε, the properties of Fε andthe Dominated Convergence Theorem, we find out that∫

Tj

|∇u(z)|pdz −∫

∂Tj

u(z)|∇u(z)|p−2∂u(z)

∂νdz ≤ [(1− t)λ0 + tλ1]

∫Tj

u(z)pdz

≤ [(1− t)λ0 + tλ1]

∫Ωt

u(z)pdz .

We may rewrite the last inequality in the following way∫Aj\Bj

|∇u(z)|pdz −∫

∂Aj∪∂Bj

u(z)|∇u(z)|p−2∂u(z)

∂νdz ≤ [(1− t)λ0 + tλ1]

∫Ωt

u(z)pdz. (32)

11

As u → 0 for z → ∂Ωt and ∇u → 0 for z → ∂Ωt (recall that Ωt = ∇v = 0 = ∇u = 0),passing to the limit for j → +∞ we have∫

Ωt\Ωt

|∇u(z)|pdz ≤ [(1− t)λ0 + tλ1]

∫Ωt

u(z)pdz. (33)

But ∫Ωt\Ωt

|∇u(z)|pdz =

∫Ωt

|∇u(z)|pdz ,

so (33) becomes ∫Ωt

|∇u(z)|pdz ≤ [(1− t)λ0 + tλ1]

∫Ωt

u(z)pdz. (34)

Finally, we get

λt ≤∫

Ωt|∇u(z)|pdz∫

Ωtu(z)pdz

≤ [(1− t)λ0 + tλ1] , (35)

i.e. (17).

Remark 3.3 Concerning equality conditions in the Brunn-Minkowski inequality for λ, weconjecture that they are the same as in the other known cases: if K0 and K1 are such thatequality occurs in (4), then they are homothetic. This claim is true for p = 2 as showed in[7]; moreover we are able to prove it in some other special cases. We start with two remarks.(i) If K0 and K1 are convex bodies in Rn, with boundary of class C2, such that there isequality in (4), we may assume, after a normalization, that λ0 = λ1 = λt = 1; indeed, fori = 0, 1, let

Ti = λ1/pi Ki , η =

tλ−1/p1

(1− t)λ−1/p0 + tλ

−1/p1

;

it is easy to check that T0, T1 and Tη = (1 − η)T0 + ηT1 still render (4) an equality andtrivially λ(T0) = λ(T1) = λ(Tη) = 1. (ii) If equality holds in (4), then, by (35), the functionu is a minimizer for the quotient in the definition of λ(Kt) and therefore it is a solution of

problem (3) in Ωt; in turn, v is a solution of (16) in Ωt \ Ωt. In view of this fact in the sequelwe will write ut and vt instead of u and v, respectively.

The argument that we use to characterize equality conditions is based on the fact thatvi is of class C2,− in a neighbourhoud of ∂Ωi, i = 0, 1. Under the assumption that Ωi is justconvex, this is proved for n = 2 in [15] (where it is also conjectured for n > 2). In order tohave the same property in higher dimension, we make the stronger assumption that K0 andK1 are of class C2,+.

Case I: n = 2. We consider v0 and v1 as in the proof of Theorem 1.1; from the previousconsiderations and from Remarks 1 and 2 in [15], we obtain that vi ∈ C2,−(Ωi\Ωi), i = 0, 1, t.

For X ∈ R2\0, let z ∈ Ωt\Ωt such that

∇vt−1(X) = z .

Here we can repeat the argument contained in the proof of Lemma 2.1 to conclude that (9)and (10) hold. Moreover, we know that the following formulas hold

∆pvt(z) = − [1 + (p− 1)|∇vt(z)|p ] , ∆pv0(x) = − [1 + (p− 1)|∇v0(x)|p ] ,

∆pv1(y) = − [1 + (p− 1)|∇v1(y)|p ] .

12

In these conditions, we can apply Lemma 4.2 in [9] and deduce

D2vt(z) = D2v0(x) = D2v1(y),

which implies

D2vt

((∇vt)

−1(X))

= D2v0

((∇v0)

−1(X))

= D2v1

((∇v1)

−1(X))

and, passing to the conjugate functions,

D2(−v0)∗(X) = D2(−v1)

∗(X), for every X ∈ R2\0 .

Hence, there exists X ∈ R2 such that, for every X ∈ R2\0,

∇(−v0)∗(X) = ∇(−v1)

∗(X) + X .

Then,Ω0\Ω0 = ∇(−v0)

∗ (R2\0

)= ∇(−v1)

∗ (R2\0

)+ X = Ω1\Ω1 + X, (36)

from which one easily obtains that

Ω0 = Ω1 + X .

Case II: K0 and K1 are of class C2,+. From [18], for i = 0, 1, we know that there existsδi > 0 such that vi ∈ C2,−(Ni), where Ni = x ∈ Ωi : dist (x, ∂Ωi) < δi. Since |∇vi| tendsto +∞ as x → ∂Ωi, for i = 0, 1, t, there exists R > 0 such that

∇vi(Ni) ⊆ X ∈ Rn : |X| > R .

Repeating the argument of the previous case, we find

D2(−v0)∗(X) = D2(−v1)

∗(X) ∀X such that |X| > R .

Therefore, for some X ∈ Rn,

∇(−v0)∗(X) = ∇(−v1)

∗(X) + X ,

for every X such that |X| > R. Now, if we put Ai = x ∈ Ωi : |∇vi(x)| ≤ R, we obtain

Ω0\A0 = Ω1\A1 + X

and thenΩ0 = Ω1 + X .

13

4 Proof of Theorem 1.2

Before proving Theorem 1.2 we need some preparatory facts. As we said in the introduction,the p-torsional rigidity of K, τ(K), is defined, for p > 1, through the following formula:

1

τ(K)= inf

∫Ω|∇w(x)|pdx[∫

Ω|w(x)|dx

]p : w ∈ W 1,p0 (Ω) ,

∫Ω

|w(x)| dx > 0

. (37)

The above problem admits a minimizer. Indeed, consider the functional

F(w) =1

p

∫Ω

|∇w(x)|pdx−∫

Ω

w(x)dx, w ∈ W 1,p0 (Ω) ; (38)

a standard variational argument ensures that F has a minimizer u ∈ W 1,p0 (Ω) and it can be

immediately seen that u provides a solution to problem (37) also; hence

τ(K) =

[∫Ω

u(x)dx]p∫

Ω|∇u(x)|pdx

. (39)

In addition, u is a weak solution of∆pu = −1 in Ωu = 0 on ∂Ω .

(40)

Applying the Gauss-Green formula we obtain∫Ω

|∇u(x)|pdx =

∫Ω

u(x)dx , (41)

and using (39) we find the following relation

τ(K) =

[∫Ω

u(x)dx

]p−1

. (42)

Remark 4.1 Concerning specific properties of the solution u of (40), let us recall thatu ∈ C1,α(K) for some α ∈ (0, 1) (cfr [21] and [22]), and it is unique for the strict convexityof the functional F. Moreover, Sakaguchi proved in [18] that u is p−1

p-concave, i.e.

v(x) := u(x)p−1

p

is a concave function.

Let K0, K1 ∈ Kn, t ∈ [0, 1] and consider Kt = (1 − t)K0 + tK1. For i = 0, 1, t, letui : Ki → R be a bounded function. The concavity function c of u0, u1 and ut is defined inK0 ×K1 by

c(x, y) = ut((1− t)x + ty)− [(1− t)u0(x) + tu1(y)] . (43)

The notion of concavity function, depending only on one function, was originally introducedby Koorevar in [14], where the so-called Korevaar concavity maximum principle is proved,

14

a tool which permits to establish the quasi-concavity of solutions of elliptic boundary-valueproblems in many important cases. Subsequently, the same technique was employed byvarious other authors; we mention in particular Kennington who improved in [13] the resultsof Korevaar. We now establish a version of concavity maximum principle for three functions.

We setc = inf c(x, y) : (x, y) ∈ K0 ×K1 .

Theorem 4.1 Let K0, K1 ∈ Kn, t ∈ [0, 1], Kt = (1 − t)K0 + tK1. For i = 0, 1, t, letui ∈ C2(Ωi), where Ωi denotes the interior of Ki and assume that ui is solution of

n∑r,s=1

ars(∇ui(x))(ui)rs(x) + b(ui(x),∇ui(x)) = 0 ,

for x ∈ Ωi, where ars(p) is a real symmetric positive semidefinite matrix, for every p ∈ Rn

and b > 0. Assume that, for every p ∈ Rn, b(·, p) is strictly decreasing and harmonic concave(i.e. (b(·, p))−1 is concave). Under these assumptions, if c < 0, then c is not attained inΩ0 × Ω1.

The proof of this result is a mere repetition (with the obvious modifications) of the one givenby Kennington in the case of one function: see Theorem 3.1 in [13].

The last preliminary result is the Prekopa-Leindler inequality.

Theorem 4.2 (Prekopa-Leindler Inequality) Let f, g, h ∈ L1(Rn) be nonnegative func-tions and t ∈ (0, 1). Assume that

h ((1− t)x + ty) ≥ f(x)1−tg(y)t ,

for all x, y ∈ Rn. Then∫Rn

h(x)dx ≥(∫

Rn

f(x)dx

)1−t (∫Rn

g(x)dx

)t

.

In addition, if equality holds then f coincides a.e. with a log-concave function and there existC ∈ R, a > 0 and y0 ∈ Rn such that

g(y) = C f(ay + y0) for almost every y ∈ Rn .

For the proof of the Prekopa-Leindler inequality we refer to [11]. The equality condition iscontained in Theorem 12 in [10].

Proof of Theorem 1.2. We are going to prove the inequality

τ(Kt) ≥ τ(K0)1−tτ(K1)

t ; (44)

notice that this inequality implies in particular

τ(Kt) ≥ minτ(K0) , τ(K1) (45)

15

and the latter implies (6) via the same argument that we have used for the Poincare constant,at the beginning of the proof of Theorem 1.1. We denote by ui the solution of (40) in Ωi,for i = 0, 1, t. The crucial part of the proof is to establish the following inequality

ut((1− t)x + ty)p−1

p ≥ (1− t)u0(x)p−1

p + tu1(y)p−1

p , ∀x ∈ Ω0, y ∈ Ω1 . (46)

Before we prove it, let us see how this leads to the conclusion. First of all, by the arithmetic-geometric mean inequality it follows

ut((1− t)x + ty) ≥ u0(x)1−tu1(y)t , ∀x ∈ Ω0, y ∈ Ω1 . (47)

Now, extend ui as zero in Rn \ Ωi, for i = 0, 1, t. Inequality (47) continues to hold; indeed,if either x /∈ Ω0 or y /∈ Ω1, then the right hand-side vanishes and the left hand-side isnonnegative. Hence we may apply the Prekopa-Leindler inequality and (44) follows.

Firstly, we prove (46) assuming that K0 and K1 have boundary of class C2. Let vi =

u(p−1)/pi , i = 0, 1, t, and consider the concavity function c : K0 ×K1 → R,

c(x, y) = vt ((1− t)x + ty)− [(1− t)v0(x) + tv1(y)] . (48)

As ui ∈ C(Ki), for i = 0, 1, t, the infimum of c is attained at some point (x, y) ∈ K0 ×K1.Once we have proved that

c = c(x, y) = minK0×K1

c(x, y) ≥ 0 ,

then we have (46). If t = 0 or t = 1 then

c(x, y) ≡ 0, for every (x, y) ∈ K0 ×K1 .

So, from now on, we assume that t ∈ (0, 1). There are three cases we have to deal with:

i) (x, y) ∈ ∂K0 × ∂K1;

ii) (x, y) ∈ (∂K0 × Ω1) ∪ (Ω0 × ∂K1);

iii) (x, y) ∈ Ω0 × Ω1.

Case i) If (x, y) ∈ ∂K0 × ∂K1, from the boundary conditions satisfied by v0 and v1, itfollows

c(x, y) = vt((1− t)x + ty) ≥ 0 .

Case ii) This case leads to a contradiction; indeed suppose that (x, y) ∈ ∂K0 × Ω1. Asx ∈ ∂K0, u0(x) = v0(x) = 0, and as K0 has boundary of class C2, we may apply the HopfLemma (see [22], Proposition 3.2.1) and obtain that

∂v0(x)

∂ν= −∞ , (49)

where ν is the outer normal unit vector to ∂K0 at x. Now, we consider

ϕ(s) = c(x + sν, y + sν), for s ∈ (−δ, 0) ,

16

where δ > 0 is sufficiently small. According to our assumption, ϕ attains its global minimumat s = 0, on the other hand (49) implies

lims→0−

ϕ′(s) = +∞ ,

these two facts are in contradiction. The case (x, y) ∈ Ω0 × ∂K1 is completely analogous.Case iii) Also in this case we argue by contradiction. Suppose that

c = c(x, y) < 0 . (50)

For η > 0 and for i = 0, 1, t we set

Ωi,η = x ∈ Ωi | dist (x, ∂Ωi) > η .

As (x, y) ∈ Ω0 ×Ω1, there exist δ > 0 such that (x, y) ∈ Ω0,δ ×Ω1,δ. Let δ′ > 0 be such that

(1− t)Ω0,δ + tΩ1,δ ⊂ Ωt,δ′ ;

in particular z = (1− t)x+ ty ∈ Ωt,δ′ . In the proof of Theorem 2 in [18] it is showed that, fori = 0, 1, we can construct a family of functions ui,ε, depending on a real parameter ε > 0,which satisfies the following properties for 0 < ε < εi(δ):

a) ui,ε ∈ C∞(Ωi,δ);

b) ui,ε converges uniformly to ui in the closure Ki,δ of Ωi,δ;

c) vi,ε := u(p−1)/pi,ε ∈ C∞(Ωi,δ) and converges uniformly to vi in Ki,δ;

d) vi,ε are solution ofn∑

r,s=1

ars(∇vi,ε)(vi,ε)rs + b(vi,ε, ∇vi,ε) = 0 ,

where b(vi,ε, ∇vi,ε) > 0, b(·, p) is strictly decreasing and harmonic concave for everyp ∈ Rn.

Clearly, the same can be done for Ωt and we obtain a sequence ut,ε having properties a) - d)for ε < εt(δ

′) and with Ωi,δ replaced by Ωt,δ′ . Let εδ := minε0(δ), ε1(δ), εt(δ′) . Now, we

introduce the function cε : K0,δ ×K1,δ → R

cε(x, y) = vt,ε ((1− t)x + ty)− [(1− t)v0,ε(x) + tv1,ε(y)] .

For 0 < ε < εδ, cε ∈ C∞(Ω0,δ × Ω1,δ) and

cε(x, y) → c(x, y) uniformly in K0,δ ×K1,δ . (51)

¿From (50) and (51) it follows that cε admits a negative global minimum, for ε sufficientlysmall; let (xε, yε) be the point where such minimum is attained and let cε = cε(xε, yε). Weapply Theorem 4.1 and we deduce that cε can not be attained in Ω0,δ × Ω1,δ. Consequently

(xε, yε) ∈ ∂ (K0,δ ×K1,δ) (52)

17

for ε sufficiently small. So

minK0,δ×K1,δ

cε(x, y) = min∂(K0,δ×K1,δ)

cε(x, y) = cε(xε, yε) .

As ε tends to 0+ we obtain that

minK0,δ×K1,δ

c(x, y) = min∂(K0,δ×K1,δ)

c(x, y) = c < 0 , (53)

and this holds for every δ sufficiently small. If we let δ tend to 0+ we have the followingresult

minK0×K1

c(x, y) = min∂(K0×K1)

c(x, y) < 0 , (54)

which is in contradiction with the previous discussion of cases i) and ii). Inequality (46) isthen proved.

Next, we remove the assumption on the regularity of ∂K0 and ∂K1. For an arbitraryK ∈ Kn, let u be the solution of (40) in Ω. There exists a sequence Ωjj∈N, with boundaryof class C2, such that

Ωj ⊂ Ωj+1,+∞⋃j=1

Ωj = Ω .

For every j ∈ N, there exists a unique function uj ∈ W 1,p0 (Ωj) such that it solves (40) in Ωj,

or equivalently, it minimizes (38) in Ωj. If we extend it in the following way

uj(x) = 0 in Ω\Ωj ,

then uj ∈ W 1,p0 (Ω). So, for the minimizing properties of u in Ω,

F(Ωj) ≥ F(Ω), ∀ j ∈ N

(see (38) for the definition of F). ¿From (41), (38)and the above inequality it follows∫Ωj

|∇uj|pdx ≤∫

Ω

|∇u|pdx ; (55)

this and the Poincare inequality imply that the sequence uj is bounded in W 1,p0 (Ω). Therefore

we can find a subsequence uj′ and a function u ∈ W 1,p0 (Ω) satisfying uj′ u in W 1,p

0 (Ω) asj′ → +∞. As the weak limit of weak solutions is still a weak solution, u is a solution of (40)in Ω and as such solution is unique u = u (in particular, the sequence uj converges to u andnot just a subsequence of it). ¿From (55) and the lower semi-continuity of

w →∫

Ω

|∇w|pdx, w ∈ W 1,p0 (Ω) ,

it follows that

limj→+∞

∫Ω

|∇uj|pdx =

∫Ω

|∇u|pdx .

18

Using this fact and the weak convergence in W 1,p0 we obtain

∇uj → ∇u in Lp(Ω)

and consequentlyuj → u in Lp(Ω) .

In particular uj converges almost everywhere in Ω, and, if we set vj(x) = uj(x)p−1

p , we havethat vj converges almost everywhere in Ω. As v and vj are concave, we have point-wiseconvergence in Ω and uniform convergence on compact subsets of Ω.

Given K0 and K1 in Kn, let Ω0,j and Ω1,j be two sequences of open sets approximatingthe interior of K0 and K1 respectively, constructed as above, and let

Ωt,j = (1− t)Ω0,j + tΩ1,j .

With obvious extension of notation, for i = 0, 1, t, let ui,j be the solution of problem (40)

in Ωi,j, and vi,j = u(p−1)/pi,j . For x ∈ Ω0,j and y ∈ Ω1,j, z = (1 − t)x + ty ∈ Ωt,j and, for the

previous part of the proof,

vt,j(z) ≥ (1− t)v0,j(x) + tv1,j(y) . (56)

As j tends to +∞, for i = 0, 1, t,ui,j → ui ,

where ui is the solution of (40) in Ωi. Letting j → +∞ we obtain (46). The proof of (6) isconcluded.

Finally, we consider the equality case. Let K0, K1 and t be such that equality holds;recalling the argument at the beginning of the proof, we have that ut, u0 and u1, extendedas zero in Rn \ Ωt, Rn \ Ω0 and Rn \ Ω1 respectively, give equality in the Prekopa-Leindlerinequality. Hence, by Proposition 4.2 we deduce that

u1(y) = C u0(ay + y0) , (57)

where C, a > 0 and y0 ∈ Rn. Since ui(x) > 0 if and only if x ∈ Ωi, i = 0, 1, we deduce thatK0 and K1 have to be homothetic.

Remark 4.2 Theorem 1.2 can be proved also in the class of compact sets with boundary ofclass C2. This fact is proved in [7] (Theorem 2.9) in the special case p = 2 but the argumentcan be adapted to the general situation p > 1.

Remark 4.3 We present an extension of Theorem 1.2. Let K ∈ Kn and let Ω denote itsinterior. For 0 ≤ α < p− 1, and p > 1, we may consider the following problem

∆pu = −uα, u > 0 in Ωu = 0 on ∂Ω .

(58)

By direct methods in the Calculus of Variations it is possible to prove the existence of asolution u of (58) as minimizer of

F(v) =1

p

∫Ω

|∇v|pdx−∫

Ω

v1+αdx

19

in W 1,p0 (Ω). Such solution is unique; this kind of result has been the subject of several papers,

see for instance [12]. Then, one can define the functional

τα(K) =

(∫Ω

|∇u|pdx

)p−1

, (59)

where u is the solution of (58). Following the lines of the proof of Theorem 1.2, it is possibleto prove that τα satisfies a Brunn-Minkowski type inequality. A similar result is proved in[7] for p = 2.

References

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[2] C. Borell, Capacitary inequalities of Brunn-Minkowski type, Math. Ann. 263 (1984),179-184.

[3] C. Borell, Hitting probability of killed Brownian motion: a study on geometric regularity,Ann. Sci. Ecole Norm. Super. Paris, 17 (1984), 451-467.

[4] C. Borell, Greenian potentials and concavity, Math. Ann. 272 (1985) 155-160.

[5] H. J. Brascamp, E. H. Lieb, On extensions of the Brunn-Minkowski and Prekopa-Leindler theorems, including inequalities for log-concave functions, and with an appli-cation to the diffusion equation, J. Funct. Anal. 22 (1976), 366-389.

[6] L. A. Caffarelli, D. Jerison, E. H. Lieb, On the case of equality in the Brunn-Minkowskiinequality for capacity, Adv. Math. 117 n.2 (1996), 193-207.

[7] A. Colesanti, Brunn-Minkowski inequalities for variational functionals and related prob-lems, preprint (2003).

[8] A. Colesanti and P. Cuoghi, The Brunn-Minkowski inequality for the n-dimensionallogarithmic capacity, preprint (2003).

[9] A. Colesanti and P. Salani, The Brunn-Minkowski inequality for p-capacity of convexbodies, to appear on Math. Ann.

[10] S. Dubuc, Criteres di convexite et inegalites integrales, Ann. Inst. Fourier, Grenoble 27,1 (1977), 135-165.

[11] R.J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39, 3(2002), 355-405.

[12] M. Belloni and B. Kawohl, A Direct Uniqueness Proof for Equations Involving the p-Laplace Operator, Manuscripta Math., 109, 229-231 (2002).

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