Brunn-Minkowski inequality for variational functional involving the P-Laplacian operator

Acta Mathematica Scientia 2009,29B(5):1143–1154 http://actams.wipm.ac.cn

BRUNN-MINKOWSKI INEQUALITY FOR VARIATIONAL FUNCTIONAL INVOLVING THE P -LAPLACIAN OPERATOR∗ Hu Huaxiang (


)

)∗∗

Zhou Shuqing (

Department of Mathematics, Hunan Normal University, Changsha 410081, China E-mail: [email protected]; [email protected]

Abstract Laplacian

In this paper, we investigate the following elliptic problem involving the P -

 −Div(|∇u|  u > 0

p−2

(P)

∇u) = |u|q−1 u

u=0

in K, in K, on ∂K,

where p > 1, 0 < q < p − 1, K ⊂ Rn with K ∈ Kn , and prove that the energy integral of the problem (P) satisfies a Brunn-Minkowski type inequality. Key words P -Laplacian; energy integral; Brunn-Minkowski type inequlity 2000 MR Subject Classification

1

35J20; 35J60

Introduction and Main Result

The intention of this paper is to investigate Brunn-Minkowski type inequality for energy integral of solution to the following elliptic problem involving the P -Laplacian ⎧ p−2 q−1 ⎪ ⎪ ⎨ −Div(|∇u| ∇u) = |u| u in K, (1) u>0 in K, ⎪ ⎪ ⎩ u=0 on ∂K, where p > 1, 0 < q < p − 1 and K is an open set in Rn with K ∈ Kn , here Kn denotes the set of convex bodies (compact convex set with nonempty interiors) in Rn . The variational functional corresponding to (1) is   1 1 F (u) = |∇u|p dx − |u|q+1 dx, (2) p K q+1 K ∗ Received

May 25, 2008; revised March 30, 2009. The Project supported by Natural Science Foundation of China (10671064), Scientific Research Fund of Hunan Provincial Education Department (06C516) and Excellent Youth Programm of Hunan Normal University (080640) ∗∗ Corresponding author: Zhou Shuqing

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where u ∈ W01,p (K). If u is a solution to (1) in K, we define the energy integral of u with respect to K by  E(K) = |∇u|p dx. (3) K

In order to state our main result, we first give some notations. Throughout this paper, we use following notations: K and Ki (i = 0, 1, t) denote open sets in Rn with the closures K, K i ∈ Kn , respectively. We also fix m = p − q − 1, where p and q are given in (1). Our goal is to prove that E(K) satisfies a Brunn-Minkowski inequality. The classical Brunn-Minkowski inequality states that, if K0 , K1 ∈ Rn are convex sets with compact closure, and t ∈ [0, 1], then 1

1

1

V n ((1 − t)K0 + tK1 ) ≥ (1 − t)V n (K0 ) + tV n (K1 ), where (1 − t)K0 + tK1 = {(1 − t)x + ty : x ∈ K0 , y ∈ K1 } and V (K) is the n-dimensional Lebesgue measure of K. The equality holds if and only if K0 is homothetic to K1 , that is, K1 = x + tK0 for some x ∈ Rn , t > 0. For more details about the Brunn-Minkowski inequality, the readers can consult the book [25] written by Schneider and the survey article [16] by Gardner. The Brunn-Minkowski inequality is one of the most and deepest results in the theory of convex bodies [25], and it is associated with other fundamental inequalities such as the isoperimetric inequality, the Sobolev inequality and the Pr´ekopa-Leindler inequality [16]. It can also be applied to other problems of convexity including convex functions, convex geometry, probability theory on convex sets, and computational complexity, see for example, in [1], [3], [4], [16], [20], [21], [25], [26], [29], [31] and references therein. In [11], Colesanti collected three main examples for Brunn-Minikowski inequalities: the Brunn-Minkowski inequality for the first eigenvalue of the Laplace operator (see details in [5], [9]), the Newtonian capacity (in [6], [7], [10], [12]) and the torsional rigidity (in [8]). The Brunn-Minkowski inequalities were extended in various directions in [12], [14], [24]. Very recently, Colesanti studied in [11] the following problem: ⎧ p ⎪ ⎪ ⎨ Δu = −u in K, ⎪ ⎪ ⎩

u>0

in K,

u=0

on ∂K,

(4)

where K ∈ Rn , 0 < p < 1, and proved that the energy integral  |∇u|2 dx F (K) = K

of the solution to (4) satisfies a Brunn-Minkowski inequality. Later, Colesanti, Cuoghi and Salani [13] proved the Poincar´e constant λ(K) of domain K ∈ Rn , defined by   p 1,p p K |∇v| dx  : v ∈ W0 (K), |v| dx > 0 , (5) λ(K) = inf |v|p dx K K verifies a Brunn-Minkowski inequality:

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Theorem A ([13]) Let K 0 , K 1 ∈ Kn with C 2 boundary and let p > 1. For t ∈ [0, 1], the following inequality holds: 1

1

1

λ− p ((1 − t)K0 + tK1 ) ≥ (1 − t)λ− p (K0 ) + tλ− p (K1 ).

(6)

Sakaguchi proved in [23] that the infimum of (5) is achieved by some function u ∈ W01,p (K), which is a weak solution of ⎧ p−2 p−2 ⎪ ⎪ ⎨ −div(|∇u| ∇u) = λ(K)|u| u in K, ⎪ ⎪ ⎩

u>0

in K,

(7)

u=0

on ∂K.  If u is normalized such that K |u|p dx = 1, then λ(K) = K |∇u|p dx, and Theorem A  means that the Brunn-Minkowski inequality holds for the energy integral λ(K) = K |∇u|p dx of the solution to (7). In this article, we make a further extension, proving a Brunn-Minkowski inequality for the energy integral of solution to a different class of equations involving p-Laplacian, and the boundary of convex bodies need not be of C 2 . Our main results can be stated as follows. Theorem The energy integral E(K) defined by (3) satisfies the Brunn-Minkowski inequality, i.e., 1 1 1 E α ((1 − t)K0 + tK1 ) ≥ (1 − t)E α (K0 ) + tE α (K1 ) (8) 

holds for all open convex sets K0 , K1 ∈ Rn with K 0 , K 1 ∈ Kn , t ∈ [0, 1] and α = n + p(q+1) m . Moreover, the equality holds if and only if K0 is homothetic to K1 . The proof of our main result is based on the homogeneity of the energy integral (3) and the concavity property of the solution to (1). To obtain the concavity property of the solution to (1), we need Korevaar’s concavity maximum principle [22] and its improved versions due to Kennington [20], Kawohl [18], [19] and Colesanti, Cuoghi and Salani [13]. But the above concavity maximum principle only works for classical solutions. Owing to the degeneracy of our problem, the solutions to (1) are only weak solutions, that is, the solutions of (1) belong to C 1+α (Ω) for some 0 < α < 1 and not always belong to C 2 (Ω) (see for example in [23], [27], [28], [30]). Inspired by Sakaguchi [23], we study the regularized problem (10) to (1), and approximate the solution to (1) by the solution of regularized problem (see details in Section 3). And also, since the boundary of convex bodies need not be of C 2 , we cannot use Sakaguchi’s method in [23] directly. We use a technique called domain approximation, i.e., approximating the convex bodies by a sequence of convex bodies with C 2 boundaries. The article is organized as follows, in Section 2, we use super-sub solution method and comparison principle to study the existence and uniqueness of positive solution to equation (1); in Section 3, we study the existence of the regularized problem (10), and approximating the solution to (1) by the solutions to regularized problem (10); in Section 4, we obtain concavity property of the solution to (1); finally, we prove our main result.

2

The Existence and Uniqueness of Positive Solution

The purpose of this section is to study the existence and uniqueness of solution to equation (1). We use the sub-super solution method and comparison principle.

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First, we consider following problem ⎧ p−2 ⎪ ⎪ ⎨ −Div(|∇u0 | ∇u0 ) = 1 in K, u0 > 0 in K, ⎪ ⎪ ⎩ u0 = 0 on ∂K.

Vol.29 Ser.B

(9)

From [17], we have Lemma 2.1 Let u0 ∈ W01,p (K) be the unique solution to (9), then there exist constants 0 < β < 1 and M0 > 0 such that u0 ∈ C β (K) and satisfies u0 C β (K) ≤ M0 . In particular, we have that u0 L∞ (K) ≤ M for some constant M > 0. Lemma 2.2 Suppose that u0 ∈ W01,p (K) is a solution to (9). Let M be the number q obtained in Lemma 2.1, and let γ ≥ M m , then γu0 is a supersolution to (1). Let φ ∈ W01,p (K) be a solution to ⎧ p−2 p−1 ⎪ in K, ⎪ ⎨ −Div(|∇φ| ∇φ) = λφ φ>0 in K, ⎪ ⎪ ⎩ φ=0 on ∂K. Since q < p − 1 and φL∞ (K) ≤ M with M as in Lemma 2.1, we can easily obtain Lemma 2.3 ξφ is a subsolution to (1) for ξ > 0 small enough. Lemma 2.4 (weak comparison principle) Let Ω be a bounded domain in Rn (n ≥ 2) with smooth boundary ∂Ω. Let u1 , u2 ∈ W 1,p (Ω) satisfy −Div(|∇u1 |p−2 ∇u1 ) ≤ −Div(|∇u2 |p−2 ∇u2 ) in Ω in weak sense. Then the inequality u1 ≤ u2 on ∂Ω implies that u1 ≤ u2 in Ω. For the proof of Lemma 2.4, we refer readers to Tolksdorf [28] (Lemma 3.1, p.800–801) or Lemma A.2 in the Appendix in Sakaguchi [23]. For u0 and φ as in Lemma 2.2 and Lemma 2.3, we can easily achieve by Lemma 2.4 that Lemma 2.5 For 0 < ξ ≤

q2 +m p−1 1 p−1 λ

M

, x ∈ K, we have

−Div(|∇(ξφ)|p−2 ∇(ξφ)) ≤ −Div(|∇(γu0 )|p−2 ∇(γu0 )), hence, ξφ ≤ γu0 . Lemma 2.6 Equation (1) admits only one solution u ∈ W01,p (K). Proof The existence of solution in W01,p (K) to (1) follows directly from the sub-super solution method in addition to Lemma 2.1, Lemma 2.3 and Lemma 2.5. The uniqueness of solution can be handled similarly as in [23]. Let u1 , u2 ∈ W01,p (K) be ¯ As in Aubin ([2] p.103), we define two positive solutions to (1), then we have u1 , u2 ∈ C01,α (K). a number b by b = sup{μ ∈ R; u1 − μu2 > 0 in Ω}. Then, similar to [23], we have b > 0. Furthermore, we may assume b ≤ 1 (otherwise, we define c by c = sup{μ ∈ R; u2 − μu1 > 0 in Ω}, then we have c ≤ 1), then since 0 < q < p − 1, we have (u1 )q ≥ (bu2 )q = bp−1 bq−p+1 (u2 )q ≥ bp−1 (u2 )q ,

No.5

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which implies the inequality (A.9) and the inequality in the last two lines on Page 419 in [23] still holds, then, we can proceed as in [23] to conclude that u1 − bu2 = 0 in Γ, where Γ is an open connected neighborhood of ∂Ω in Ω. Hence u2 and bu2 are two solutions of (1) in Γ, which implies b = 1. By the definition of b, we have u1 ≥ u2 in Ω. Similarly, we have u2 ≥ u1 in Ω. Hence we obtain u2 = u1 in Ω, this shows (1) admits at most one positive solution u ∈ W01,p (K). Remark 2.7 The existence of positive solutions to (1) can also be obtained by considering the minimizer to (2), and by the uniqueness, we see they are the same.

3 Approximation to the Solution by the Solutions of Regularized Problems In this section, we study the existence for regularized problem of (1): ⎧ 2(q+1) 2(q+1) 2q+2−p ε(q+1) 2 p−2 2 p−2 p p p ⎪ 2 ∇u] = uq − 2 u −div[(εu + |∇u| ) (εu + |∇u| ) ⎪ p ⎨ u>0 ⎪ ⎪ ⎩ u=0 for sufficiently small ε > 0. The functional corresponding to equation (10) is   2(q+1) p 1 1 Fε (u) = (|∇u|2 + ε|u| p ) 2 dx − |u|q+1 dx. p Ω q+1 Ω

in K, in K,

(10)

on ∂K,

(11)

We approximate the solution of (1) by the solutions of regularized problems (10) or (11). Concerning problem (10) or the minimizer of (11), we have Lemma 3.1 Let uε ∈ W01,p (K) be a solution to (10), then uε ∈ C β (K) for some β (0 < β < 1) and satisfies uε C β (K) ≤ M, where M and β are constants independent of ε. Lemma 3.2 There exists at least one minimizer uε ∈ W01,p (K) for Fε (u) satisfying uε > 0 in K, and moreover, uε → u in W01,p (K) as ε → 0, where u is the unique solution of (1). Lemma 3.3 For any δ > 0, there exist ε0 > 0 and T > 0 (sufficiently small) such that, for any 0 < ε < ε0 , we have M ≥ uε ≥ T > 0 in Kδ , where Kδ = {x ∈ K|dist(x, ∂K) > δ} and M is the constant in Lemma 3.1. And the minimizer uε is a solution to the Euler-equation (10). Lemma 3.4 For any δ > 0 and any 0 < ε < ε0 , the solution uε to (10) belongs to C ∞ (Kδ ) and satisfies uε C 1+β (K 2δ ) ≤ C, where β (0 < β < 1) and C are constants independent of ε, and ε0 is the constant obtained in Lemma 3.3. The proofs of Lemma 3.1–Lemma 3.4 are similar to those of Proposition 6.1, Proposition 6.2, Proposition 4.1, and Proposition 4.2 in Sakaguchi [23] with small changes. We omit them here. For the proof of Lemma 3.4, we also refer readers to Tolksdorf [27].

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Using Arzel` a-Ascoli’s theorem, we obtain from Lemma 3.1 and Lemma 3.2 that Lemma 3.5 uε → u uniformly in K as ε → 0, where u is the unique solution of (1). In view of Lemma 3.3 and Lemma 3.4, we obtain m Lemma 3.6 For 0 < ε < ε0 , define vε = uεp , then vε ∈ C ∞ (Kδ ) and converges uniformly to some v in K δ . We recall the fact that a function f (x) is called harmonic concave if and only if f −1 (x) is concave with respect to x. Lemma 3.7 vε (defined in Lemma 3.6) is a solution to equation n

ar,s (∇vε )Dr,s vε + b(vε , ∇vε ) = 0 in Kδ ,

(12)

r,s=1

where

n

r,s=1

 ar,s (∇vε )Dr,s vε = div

p2 ε + 2 |∇vε |2 m

p−2 2

 ∇vε ,

  

p−2 p2 (p − 1)(q + 1) m 1 mε(q + 1) 2 2 2 b(vε , ∇vε ) = |∇vε | − , ε + 2 |∇vε | + vε m m p2 p here, (ar,s (q)) is a real symmetric positive definite matrix. If we choose ε1 small enough, depending on p, q, then for ε ∈ (0, min{ε0 , ε1 }), b(., q) is striclty decreasing and harmonic concave for every q ∈ Rn . m Proof Since vε = uεp and (10), it is easy to prove that vε satisfies (12). It suffice to prove the rest of them. By simple calculation, we obtain   n

p−2

p2 2 2 ar,s (∇vε )Dr,s vε = div ε + 2 |∇vε | ∇vε m r,s=1  2 n

p−4

 ∂2v

p (p − 2) ∂vε ∂vε p2 p2 2 ε 2 2 = ε + 2 |∇vε | · + ε + 2 |∇vε | δrs . 2 m m ∂x ∂x m ∂x ∂x r s r s r,s=1 

∂vε ∂xr

2

In order to prove that (ar,s (∇vε )Dr,s vε ) is positive definite, we need to prove A = ( p p2 ∂vε 2 · ∂x + (ε + m 2 |∇vε | )δrs ) is positive definite. s For any ξ ∈ Rn \ {0}, ξ = (ξ1 , · · · , ξn ),  n 

n



p2 (p − 2) ∂vε ∂vε p2  ⊥ 2 S = ξAξ = ξ · ξ⊥ · ξ · ξ + ε + |∇v | r s ε m2 ∂xr ∂xs m2 s=1 r=1  n 

 n

∂vε p2 p2 (p − 2) ∂vε 2 + ε + |ξ|2 = · ξ · ξ |∇v | r s ε 2 m2 ∂x ∂x m r s r=1 s=1

p2 (p − 2) p2 2 2 |ξ|2 . = [ξ · ∇v ] + ε + |∇v | ε ε m2 m2 It is obvious that S > 0 when p ≥ 2. When 1 < p < 2, we can estimate S as follows. Since p − 2 < 0 and [ξ · ∇vε ]2 ≤ |∇vε |2 |ξ|2 , we have p2 (p − 2) p2 (p − 2) 2 [ξ · ∇v ] ≥ |∇vε |2 |ξ|2 , ε m2 m2

(p−2) m2

Hu & Zhou: BRUNN-MINKOWSKI INEQUALITY INVOLVING P -LAPLACIAN OPERATOR 1149

No.5

and hence,

p2 (p − 2) p2 2 2 |ξ|2 [ξ · ∇v ] + ε + |∇v | ε ε m2 m2

p2 (p − 2) p2 2 2 2 |ξ|2 ≥ |∇v | |ξ| + ε + |∇v | ε ε 2 2 m m  2  p (p − 1) 2 ≥ |∇v | + ε |ξ|2 > 0. ε m2

S=

(13)

In a word, S > 0 for all p > 1. In the following, we need to prove b(., q) is decreasing and harmonic concave for every q ∈ Rn . Put b(vε , ∇vε ) = v1ε d(∇vε ), where  p−2    2 (p − 1)(q + 1) m p2 mε(q + 1) 2 |∇vε |2 − + . d(∇vε ) = ε + 2 |∇vε | m m p2 p Then we see that d(∇vε ) is positive for small ε. Indeed, if |∇vε |2 ≥ m p

2

> 0. And if |∇vε | <

εm2 p2 (p−1) ,

p 2

we get d(∇vε ) ≥ −Cp ε + p2 2 m2 |∇vε |

m p

εm2 p2 (p−1) ,

we get d(∇vε ) ≥

for some positive constant Cp

p p−1 ε.

depending only on p, since ε ≤ ε + ≤ Thus, we can choose a positive number ε1 depending on p, q, and we see that b(vε , ∇vε ) vε ∂b ε) is positive for 0 < ε < min{ε0 , ε1 }. Since vε is positive, 1b = d(∇v , ∂v = − d(∇v < 0 and v2 ε) ε ε

∂2 1 ∂vε2 ( b )

= 0, we obtain that b(vε , ∇vε ) is strictly positive and harmonic concave. The proof is completed.

4

The Concavity Property of the Solution

In this section, we first study the positive homogeneous order of E(K) for the positive solution of (1) or minimizer of (2). Next, we study the concavity property for the positive solution of (1) or minimizer of (2). We say that E(K) is positively homogeneous of order α if E(rK) = rα E(K) holds for all r > 0. Theorem 4.1 The energy integral E(K) is positively homogeneous of order α = n + p(q+1) m . p Proof Let u(y) be the unique solution of (1) in K, then the function v(y) = r m u( yr ), y ∈ rK, is the unique solution of (1) in rK. Hence,   p y E(rK) = |∇y v|p dy = |∇y (r m u( ))|p dy r rK rK p2 |∇x u(x)|p r−p+ m +n dx = rα E(K). (14) = K

The proof is completed. For K 0 , K 1 ∈ Kn , t ∈ [0, 1], define K t = (1 − t)K 0 + tK 1 . For i = 0, 1, t, let ui : K i → R be bounded functions, and define function C : K 0 × K 1 → R by C(x, y) = ut ((1 − t)x + ty) − [(1 − t)u0 (x) + tu1 (y)], (x, y) ∈ K 0 × K 1 .

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Also, we set C = inf{C(x, y) : (x, y) ∈ K0 × K1 }. The following Lemma 4.2 is due to Colesanti, Cuoghi and Salani (Theorem 4.1 in [13]), which originate from Kennington [20] and Korevaar [22]. Lemma 4.2 ([13]) Let K 0 , K 1 ∈ Kn , t ∈ [0, 1], Kt = (1 − t)K0 + tK1 . For i = 0, 1, t, let ui ∈ C 2 (Ki ) be the solution of n

ar,s (∇ui )(ui )r,s + b(ui , ∇ui ) = 0,

r,s=1

where ars (p) is a real symmetric positive matrix for every p ∈ Rn , and b > 0. Assume that, for every p ∈ Rn , b(., p) is strictly decreasing and harmonic concave. If C < 0, then C can not be attained in K0 × K1 . Lemma 4.3 Let K 0 , K 1 ∈ Kn , t ∈ [0, 1], and K t = (1 − t)K 0 + tK 1 . Suppose that ui are solutions to (1) in K i , i = 0, 1, t. Then m

m

m

[ut ((1 − t)x + ty)] p ≥ (1 − t)[u0 (x)] p + t[u1 (y)] p , ∀x ∈ K 0 , y ∈ K 1 .

(15) m

Proof First, we assume that the boundaries of K0 and K1 are of C 2 . Let vi = [ui (x)] p , i = 0, 1, t, and consider the function C : K 0 × K 1 → R given by C(x, y) = vt ((1 − t)x + ty) − [(1 − t)v0 (x) + tv1 (y)]. Because ui ∈ C(K i ) for i = 0, 1, t, the infimum of C is achieved at some point (x, y) ∈ K 0 × K 1 . Once we have proved that C = C(x, y) = min C(x, y) ≥ 0, then we have (15). We may assume K 0 ×K 1

t ∈ (0, 1) and have to consider three cases. Case I. (x, y) ∈ ∂K0 × ∂K1 . In this case, v0 (x) = v1 (y) = 0, therefore C(x, y) = vt ((1 − t)x + ty) ≥ 0.  Case II. (x, y) ∈ (∂K0 × K1 ) (K0 × ∂K1 ). This is impossible. In fact, without loss of generality, we may suppose that (x, y) ∈ ∂K0 × K1 . Then u0 (x) = v0 (x) = 0, and since K0 has C 2 boundary, Hopf Lemma implies that ∂v0 (x) = −∞, ∂ν

(16)

where ν is the outer unit normal vector to ∂K0 at x. For δ > 0 sufficiently small and s ∈ (−δ, 0), we define φ(s) = C(x + sν, y + sν), then φ attains its global minimum at s = 0, and φ (0) ≤ 0. On the other hand, (16) implies lim− φ (s) = +∞. That is a contradiction. s→0

Case III. (x, y) ∈ K0 × K1 . In this case, we also argue by contradiction. Suppose that C(x, y) < 0.

(17)

For η > 0 and i = 0, 1, t, we denote Kt = (1 − t)K0 + tK1 , Ki,η = {x ∈ Ki |dist(x, ∂Ki ) > η}. Since (x, y) ∈ K0 × K1 , there exits a δ > 0 such that (x, y) ∈ K0,δ × K1,δ . Let δ  > 0 such that (1 − t)K0,δ + tK1,δ ⊂ Kt,δ . In particular, z = (1 − t)x + ty ∈ Kt,δ . In Section 3, it was shown that, for i = 0, 1, we can construct a family of functions ui,ε , depending on a real parameter ε > 0, which possesses following properties for 0 < ε < εi (δ): (a) ui,ε ∈ C ∞ (Ki,δ );

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(b)

ui,ε converges uniformly to ui in K i,δ ; m

p ∈ C ∞ (Ki,δ ) and converges uniformly to vi in K i,δ ; vi,ε = ui,ε n  ar,s (∇vi,ε )(vi,ε )r,s + b(vi,ε , ∇vi,ε ) = 0, (d) vi,ε are solutions of

(c)

r,s=1

where b(vi,ε , ∇vi,ε ) > 0, b(., p) is strictly decreasing and harmonic concave for every p ∈ Rn . Clearly, the same can be done for Kt and we obtain a sequence ut,ε with properties (a)–(d) for ε < εt (δ  ) and with Ki,δ replaced by Kt,δ . Let εδ = min{ε0 (δ), ε1 (δ), εt (δ  )}. Now, we introduce the function Cε : K0,δ × K1,δ → R given by Cε (x, y) = vt,ε ((1 − t)x + ty) − [(1 − t)v0,ε (x) + tv1,ε (y)]. For 0 < ε < εδ , Cε ∈ C ∞ (K0,δ × K1,δ ) and Cε (x, y) → C(x, y) uniformly in K 0,δ × K 1,δ .

(18)

It follows from (17) and (18) that, if ε is small enough, then Cε admits a negative minimum. Let (xε , y ε ) be the minimum point of Cε and let C ε = Cε (xε , y ε ). We apply Lemma 4.2 and Lemma 3.7 in Section 3 to vi,ε to deduce that C ε cannot be attained in K0,δ × K1,δ . Consequently, (xε , y ε ) ∈ ∂(K0,δ × K1,δ ) for ε sufficiently small. So, Cε (x, y) =

min

K 0,δ ×K 1,δ

min

∂(K0,δ ×K1,δ )

Cε (x, y) = Cε (xε , yε ).

As ε tends to 0+ , we obtain min

K 0,δ ×K 1,δ

C(x, y) =

min

∂(K0,δ ×K1,δ )

C(x, y) = C < 0,

(19)

and this holds for every δ sufficiently small. If we let δ tend to 0+ , we have min C(x, y) =

K 0 ×K 1

min

∂(K0 ×K1 )

C(x, y) < 0,

which contradicts with the previous discussions of cases I and II. Inequality (15) is then proved. Next, we remove the assumption on the regularity of ∂K0 and ∂K1 . For an arbitrary K ⊂ Rn with K ∈ Kn , let u be the solution to (1) in K. There exists a sequence {Ωj }j∈N with +∞  Ωj = K (see [13], [25]). C 2 boundary such that Ωj ⊂ Ωj+1 , j=1

For every j ∈ N , there exists a unique uj ∈ W01,p (Ωj ) such that it solves (1) in W01,p (Ωj ), or equivalently, it minimizes (2) in W01,p (Ωj ). Set uj (x) = 0, x ∈ K \ Ωj , then uj ∈ W01,p (K). So, it follows from the minimizing properties of u in K that F (uj ) ≥ F (u), ∀j ∈ N . Then, it follows from (1), Gauss-Green formula and the definition of F , that  −m F (uj ) = |∇uj |p dx, p(q + 1) Ωj since m > 0, so that



 |∇uj | dx ≤

|∇u|p dx.

p

Ωj

(20)

K

This and Poincar´e inequality imply that the sequence uj is bounded in W01,p (K), therefore, we ∼ ∼ can find a subsequence uj  and a function u∈ W01,p (K) satisfying uj  u weakly in W01,p (K)

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∼

∼

as j  → +∞. In particular, u must be a solution to (1) in K and then u= u, this implies that the sequence uj  conveges to u. From (20) and the weakly lower-semi-continuity of w →  |∇w|p dx, w ∈ W01,p (K), it follows that K   lim |∇uj |p dx = |∇u|p dx. j→+∞

Ωj

K

We deduce from this fact and the weak convergence of uj in W01,p (K) that ∇uj → ∇u in Lp (K). m

Consequently, uj → u in Lp (K). In particular, uj → u a.e. in K, and, if we set vj (x) = ujp (x), we see that vj converges to some v a.e. in K. Since v and vj are concave, we have point-wise convergence in K and uniform convergence on compact subsets of K. Given K0 , K1 ⊂ Rn , let {Ω0,j } and {Ω1,j } be two sequences of open sets, constructed as above, approximating to K0 and K1 respectively. Let Ωt,j = (1 − t)Ω0,j + tΩ1,j with obvious extension of notation, for i = 0, 1, t, and let ui,j be the solution to (1) in Ωi,j , and m

p (x). From the previous part of the proof, we obtain vi,j = ui,j

vt,j ((1 − t)x + ty) ≥ (1 − t)v0,j (x) + tv1,j (y), ∀x ∈ Ω0,j , y ∈ Ω1,j .

(21)

Then, since ui,j → ui as j tends to +∞, for i = 0, 1, t, where ui is the solution to (1) in Ki , we obtain (15).

5

Proof of the Main Theorem First, we consider the multiplicative form of the inequality contained in Theorem: E((1 − t)K0 + tK1 ) ≥ E(K0 )1−t E(K1 )t , ∀K0 , K1 ∈ Kn .

(22)

We remark that, for arbitrary K0 , K1 ∈ Kn and t ∈ [0, 1], (8) follows from Theorem 4.1 and (22) by setting K0

1 −α

= (E(K0 ))

K0 , K1

1 −α

= (E(K1 ))

1



K1 , t =

t(E(K1 )) α 1

1

(1 − t)(E(K0 )) α + t(E(K1 )) α

.

Moreover, if K0 , K1 ∈ Kn and t ∈ [0, 1] render (8) to be an equality, then K0 , K1 and t defined as above give equality in (22). Hence, it suffices to prove (22) and to characterize the corresponding equality conditions. For i = 0, 1, t, let ui be the solution of problem (1) for K = Ki , we have, by divergence theorem,  E(Ki ) = |∇u|p−2 ∇u · ∇udx K   i ∂u ds − = |∇u|p−2 u · u · div(|∇u|p−2 ∇u)dx ∂n ∂K Ki  i q+1 = u dx. Ki

Hu & Zhou: BRUNN-MINKOWSKI INEQUALITY INVOLVING P -LAPLACIAN OPERATOR 1153

No.5

Let x ∈ K0 , y ∈ K1 , and z = (1 − t)x + ty ∈ Kt . From Lemma 4.3, we know that m

m

m

[ut (z)] p ≥ (1 − t)[u0 (x)] p + t[u1 (y)] p . q+1 Let us extend ui to be zero outside of Ki , i = 0, 1, t, and define f = uq+1 0 , g = u1 , h = then we have, for x ∈ K0 and y ∈ K1 ,

uq+1 , t

1

h((1 − t)x + ty) ≥ [(1 − t)f r (x) + tg r (y)] r , r =

m . p(q + 1)

By the arithmetic-geometric mean inequality, h((1 − t)x + ty) ≥ f 1−t (x)g t (y), ∀x ∈ K0 , y ∈ K1 . In fact, this inequality holds for all x, y ∈ Rn . Indeed, if either x ∈ / K0 or y ∈ / K1 , then the right-hand side vanishes. To proceed, we need a famous inequality called Pr´ekopa-Leindler inequality. Lemma 5.1 (Pr´ekopa-Leindler inequality) Let f, g, h ∈ L1 (Rn ) be nonnegative functions and t ∈ (0, 1). Assume that h((1 − t)x + ty) ≥ f 1−t (x)g t (y), ∀x, y ∈ Rn . Then  

1−t 

t h(x)dx ≥ f (x)dx g(x)dx . Rn

Rn

Rn

In addition, if the equality holds, then f coincides a.e. with a log-concave function w (i.e., log w is concave) and there exist C ∈ R, a > 0 and y0 ∈ Rn such that g(y) = Cf (ay + y0 ) for a.e. y ∈ Rn . For the proof of Lemma 5.1, we refer readers to [16]. The equality condition is contained in Theorem 12 in [15]. By making use of Lemma 5.1, we obtain  

1−t 

t h(x)dx ≥ f (x)dx g(y)dx , (23) Rn

Rn

Rn

i.e., (22) holds. Moreover, if the equality holds in (22), then (23) becomes equality and, in particular, f and g render the Pr´ekopa-Leindler inequality to be an equality. By Lemma 5.1 and the fact that f (x) is positive if and only if x ∈ K0 , and g(y) is positive if and only if y ∈ K1 , we conclude that K0 and K1 coincide up to a translation. The Theorem is completely proved. Acknowledgements The authors would like to thank professor Dai Qiuyi for useful discussion. The second author also would like to thank Chern Institute of Mathematics of Nankai University for hospitalities he received during his stay there. References [1] Agueh M, Ghoussoub N, Kang X. Geometric inequalities via a general comparison principle for inerating gases. Geom Funct Anal, 2004, 14: 215–244 [2] Aubin T. Nonlinear Analysis on Manifolds, Monge-Amp`ere Equations. New York, Heidelberg, Berlin: Springer-Verlag, 1982 [3] Bobkov S G, Ledoux M. From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom Funct Anal, 2000, 10: 1028–1052

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