Faber-Krahn inequality for anisotropic eigenvalue problems with Robin boundary conditions
aa r X i v : . [ m a t h . A P ] N ov FABER-KRAHN INEQUALITY FOR ANISOTROPIC EIGENVALUEPROBLEMS WITH ROBIN BOUNDARY CONDITIONS
FRANCESCO DELLA PIETRA AND NUNZIA GAVITONE
Abstract.
In this paper we study the main properties of the first eigenvalue λ (Ω) andits eigenfunctions of a class of highly nonlinear elliptic operators in a bounded Lipschitzdomain Ω ⊂ R n , assuming a Robin boundary condition. Moreover, we prove a Faber-Krahninequality for λ (Ω). Introduction
Let Ω be a bounded Lipschitz domain in R n , n ≥
2. This paper is devoted to the studyof the following problem:(1.1) λ (Ω) = min u ∈ W ,p (Ω) u =0 J ( u ) , where(1.2) J ( u ) = Z Ω [ H ( Du )] p dx + β Z ∂ Ω | u | p H ( ν ) dσ Z Ω | u | p dx , < p < + ∞ , ν is the outer normal to ∂ Ω, and β is a fixed positive number. Moreover,we suppose that H is a sufficiently smooth norm of R n (see Sections 2 and 3 for the preciseassumptions). The minimizers of (1.1) satisfy the equation(1.3) − div (cid:0) [ H ( Du )] p − H ξ ( Du ) (cid:1) = λ (Ω) | u | p − u in Ω , with Robin conditions on the boundary:(1.4) [ H ( Du )] p − H ξ ( Du ) · ν + βH ( ν ) | u | p − u = 0 on ∂ Ω . The operator in (1.3) reduces to the p -Laplacian when H is the Euclidean norm of R n . Fora general norm H , it is an anisotropic, highly nonlinear operator, and it has attracted anincreasing interest in last years. We refer, for example, to [1, 20, 24] ( p = 2) and [4, 6, 19, 22](1 < p < + ∞ ) where Dirichlet boundary conditions are considered. Moreover, for Neumannboundary values see, for instance, [18,36] ( p = 2), while overdetermined problems are studiedin [12,35] ( p = 2). In this paper we are interested in considering the eigenvalue problem (1.3)with the Robin boundary conditions (1.4). In particular, our main objective is to obtain aFaber-Krahn inequality by studying the shape optimization problem(1.5) min | Ω | = m λ (Ω)among all the Lipschitz domains with given measure m >
0. To study problem (1.5),we first have to investigate the basic properties of the first eigenvalue and of the relativeeigenfunctions of (1.3),(1.4), as existence, sign, simplicity and regularity.
Date : August 2, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Eigenvalue problems, nonlinear elliptic equations, Faber-Krahn inequality,Wulffshape.
In the Euclidean case, problem (1.1) reduces to λ , E (Ω) = min u ∈ W ,p (Ω) u =0 Z Ω | Du | p dx + β Z ∂ Ω | u | p dσ Z Ω | u | p dx , and the minimizers satisfy the problem − div (cid:0) | Du | p − Du (cid:1) = λ , E (Ω) | u | p − u in Ω , | Du | p − ∂u∂ν + β | u | p − u = 0 on ∂ Ω . In such a case, problem (1.5) has been first investigated by Bossel for p = 2, when Ω variesamong smooth domains of R with fixed measure. More precisely, in [7] she proved that(1.6) λ , E (Ω) ≥ λ , E ( B ) , where B is a disk such that | B | = | Ω | . This result has been generalized to any dimension n ≥ < p < + ∞ , the inequality (1.6)has been proved by [16] for smooth domains, and by [9] in the case of Lipschitz domains.The equality cases are also addressed in [9, 16]. As regards the case β <
0, we refer thereader to [25] and the references therein.In the anisotropic case, our result reads as follows. Let H o be the polar function of H ,and denote by W R the Wulff shape, that is the R -sublevel set of H o , such that |W R | = | Ω | (see Section 2 for the definitions). If Ω = W R is a Lipschitz set of R n , then λ (Ω) > λ ( W R ) . Hence, the unique minimizer of (1.5) is the Wulff shape. Such result relies in the so-calledanisotropic isoperimetric inequality (see for example [1]), and it is in agreement with theFaber-Krahn inequality for the first eigenvalue of (1.3) in the homogeneous Dirichlet case(see [4]).As a matter of fact, we may ask if the first eigenvalue λ (Ω) is bounded from above interms of the Lebesgue measure of Ω. Indeed, in the Euclidean setting, this is the case forthe first nonvanishing Neumann Laplacian eigenvalue (see [37], and also [8, 11] for relatedresults), but this does not happen for the first Dirichlet Laplacian eigenvalue. In this orderof ideas, by a result given in [29] it follows that the first Robin Laplacian eigenvalue amongthe sets of fixed measure is unbounded from above. Here we prove a lower bound for thefirst eigenvalue λ (Ω) of our anisotropic Robin problem in a convex set λ (Ω) in terms of theanisotropic inradius of Ω. This will imply that, among all Lipschitz sets with fixed measure m >
0, sup | Ω | = m λ (Ω) = + ∞ . The paper is organized as follows. In Section 2, we recall some basic definitions andproperties of H and of its polar function H o . In Section 3, we state and prove some propertiesof the first eigenvalue of (1.3), (1.4). More precisely, under suitable assumptions on H , weshow that there exists a first eigenvalue λ (Ω) which is simple. Moreover, we prove that thefirst eigenfunctions are in C ,α (Ω) ∩ C ( ¯Ω), for some 0 < α <
1. Furthermore, a solution ofthe eigenvalue problem is a first eigenfunction if and only if it has a fixed sign. In Section 4we investigate the eigenvalue problem when Ω is a Wulff shape, while in Section 5 we give arepresentation formula for λ (Ω) by means of the level sets of the first eigenfunctions. Usingsuch results, in Section 6 we state precisely the main result and give a proof. ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 3 Notation and preliminaries
Let H : R n → [0 , + ∞ [, n ≥
2, be a C ( R n \ { } ) function such that(2.1) H ( tξ ) = | t | H ( ξ ) , ∀ ξ ∈ R n , ∀ t ∈ R , and such that any level set { ξ ∈ R n : H ( ξ ) ≤ t } , with t > a ≤ b such that(2.2) a | ξ | ≤ H ( ξ ) ≤ b | ξ | , ∀ ξ ∈ R n . Remark 2.1.
We stress that the homogeneity of H and the convexity of its level sets implythe convexity of H . Indeed, by (2.1), it is sufficient to show that, for any ξ , ξ ∈ R n \ { } ,(2.3) H ( ξ + ξ ) ≤ H ( ξ ) + H ( ξ ) . By the convexity of the level sets, we have H (cid:18) ξ H ( ξ ) + H ( ξ ) + ξ H ( ξ )+ H ( ξ ) (cid:19) == H (cid:18) H ( ξ ) H ( ξ ) + H ( ξ ) ξ H ( ξ ) + H ( ξ ) H ( ξ ) + H ( ξ ) ξ H ( ξ ) (cid:19) ≤ , and by (2.1) we get (2.3).We define the polar function H o : R n → [0 , + ∞ [ of H as H o ( v ) = sup ξ =0 ξ · vH ( ξ ) . It is easy to verify that also H o is a convex function which satisfies properties (2.1) and (2.2).Furthermore,(2.4) H ( v ) = sup ξ =0 ξ · vH o ( ξ ) . The set W = { ξ ∈ R n : H o ( ξ ) < } is the so-called Wulff shape centered at the origin. We put κ n = |W| , where |W| denotes theLebesgue measure of W . More generally, we denote with W r ( x ) the set r W + x , that isthe Wulff shape centered at x with measure κ n r n , and W r (0) = W r .The following properties of H and H o hold true (see for example [3]): H ξ ( ξ ) · ξ = H ( ξ ) , H oξ ( ξ ) · ξ = H o ( ξ ) , (2.5) H ( H oξ ( ξ )) = H o ( H ξ ( ξ )) = 1 , ∀ ξ ∈ R n \ { } , (2.6) H o ( ξ ) H ξ ( H oξ ( ξ )) = H ( ξ ) H oξ ( H ξ ( ξ )) = ξ, ∀ ξ ∈ R n \ { } . (2.7) Definition 2.1 (Anisotropic area functional and perimeter ( [2, 10])) . Let M be an oriented( n − R n . The anisotropic area functional of M is σ H ( M ) := Z M H ( ν ) dσ, where ν denotes the outer normal to M and σ is the ( n − M is finite if and only if the usual Euclidean hypersurfacearea σ ( M ) is finite. Indeed, by property (2.2) we have that α σ ( M ) ≤ σ H ( M ) ≤ γ σ ( M ) . F. DELLA PIETRA, N. GAVITONE
An isoperimetric inequality for the anisotropic area holds, namely for K ⊂ M open set of R n with Lipschitz boundary,(2.8) σ H ( ∂K ) ≥ nκ n n | K | − n , and the equality holds if and only if K is homothetic to a Wulff shape (see for example [10],[15], [26], [1]). We stress that in [21] an isoperimetric inequality for the anisotropic relativeperimeter in the plane is studied.Let Ω be a bounded open set of R n , and d H ( x ) the anisotropic distance of a point x ∈ Ωto the boundary ∂ Ω , that is(2.9) d H ( x ) = inf y ∈ ∂ Ω H o ( x − y ) . By the property (2.6), the distance function d H ( x ) satisfies(2.10) H ( Dd H ( x )) = 1 . Finally, we recall that when Ω is convex d H ( x ) is concave. In a natural way, the anisotropicinradius of a convex, bounded open set Ω is the value(2.11) R H, Ω = sup { d H ( x ) , x ∈ Ω } For further properties of the anisotropic distance function we refer the reader to [13].3.
The first eigenvalue problem
In this section we prove some properties of the minimizers of (1.1), which are the weaksolutions of the following Robin boundary value problem:(3.1) ( − div ( F p ( Du )) = λ (Ω) | u | p − u in Ω ,F p ( Du ) · ν + βH ( ν ) | u | p − u = 0 on ∂ Ω . where F p ( Du ) := [ H ( Du )] p − H ξ ( Du ) . For weak solution of problem (3.1) we mean a function u ∈ W ,p (Ω) such that(3.2) Z Ω F p ( Du ) · Dψ dx + β Z ∂ Ω u p − ψ H ( ν ) dσ = λ (Ω) Z Ω | u | p − u ψ dx, ψ ∈ W ,p (Ω) . Obviously, λ (Ω) in (1.1) (and then in (3.1)) depends also on β . In general, we willconsider β > β we will denote the first eigenvalue of (3.1) with λ (Ω , β ).For the Euclidean case we refer to [31], where the eigenvalue problem for the p -Laplacianunder several boundary conditions is considered.From now on, we assume that H is a convex function as in Section 2, assuming also thatit verifies the following hypothesis:(3.3) H ∈ C ( R n \ { } ) , with n X i,j =1 ∂∂ξ j (cid:0) [ H ( η )] p − H ξ i ( η ) (cid:1) ξ i ξ j ≥ γ | η | p − | ξ | , for some positive constant γ , for any η ∈ R n \ { } and for any ξ ∈ R n . Theorem 3.1.
There exists a function u p ∈ C ,α (Ω) ∩ C ( ¯Ω) which realizes the minimum in (1.1) , and satisfies the problem (3.1) . Moreover, λ (Ω) is the first eigenvalue of (3.1) , andthe first eigenfuctions are positive (or negative) in Ω . ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 5
Proof.
The proof makes use of standard arguments. We briefly recall the main steps. Thedirect method of the Calculus of Variations guarantees that the infimum in (1.1) is attainedat a function u p ∈ W ,p (Ω). We may assume that u p ≥
0, being also | u p | a minimizerin (1.1). Moreover, the function u p is a weak solution of (3.1). In order to obtain that u p ∈ C ,α (Ω) ∩ C ( ¯Ω), we first claim that a L ∞ -estimate for u p holds. To get the claim, we take ϕ = [ T M ( u p )] kp +1 as test function, with k, M positive numbers, and T M ( s ) = min { s, M } , s ≥
0. Using (2.5) and (2.2), we easily get α ( kp + 1) Z u p ≤ M | Du p | p u kpp dx ≤≤ Z Ω F p ( Du p ) · Dϕ dx + β Z ∂ Ω u p − p ϕ H ( ν ) dσ ≤≤ λ (Ω) Z Ω u p ( k +1) p dx, and then Z Ω (cid:12)(cid:12) D T M ( u p ) k +1 (cid:12)(cid:12) p dx + Z Ω [ T M ( u p )] p ( k +1) dx ≤ (cid:18) ( k + 1) p α ( kp + 1) λ (Ω) + 1 (cid:19) Z Ω u p ( k +1) p dx. Applying the Sobolev inequality and the Fatou lemma, we get that k u p k ( k +1) p ∗ ≤ S k +1 (cid:18) ( k + 1) p kp + 1 λ (Ω) α + 1 (cid:19) p ( k +1) k u p k ( k +1) p , where S is the Sobolev constant. Using the standard Moser iteration technique for the L p -norms, we get the claim. For sake of completeness, we give the complete proof (see also [27]).First of all, we have that there exists a constant c independent of k such that (cid:18) ( k + 1) p kp + 1 λ (Ω) α + 1 (cid:19) p √ k +1 ≤ c. Then,(3.4) k u p k ( k +1) p ∗ ≤ S k +1 c √ k +1 k u p k ( k +1) p . Choosing k n in (3.4) such that ( k + 1) p = p ∗ , and k n , n ≥
2, such that ( k n + 1) p =( k n − + 1) p ∗ , by induction we obtain k u p k ( k n +1) p ∗ ≤ S kn +1 c √ kn +1 k u p k ( k n − +1) p ∗ . Hence, using iteratively the above inequality, we get k u p k ( k n +1) p ∗ ≤ S P ni =1 1 ki +1 c P ni =1 1 √ ki +1 k u p k p ∗ . Being k n + 1 = ( p ∗ /p ) n , and p ∗ /p >
1, it follows that for any n ≥ k u p k ( k n +1) p ∗ ≤ C k u k p ∗ , as r n = ( k n + 1) p ∗ → + ∞ as n → + ∞ . The estimates in (3.5) imply that u ∈ L ∞ (Ω).Indeed, if by contradiction the exist ε > A ⊂ Ω with positive measure such that | u | > C k u k p ∗ + ε = K in A , we havelim inf n k u k r n ≥ lim inf n (cid:18)Z A K r n (cid:19) rn = K > C k u k p ∗ , which is in contrast with (3.5).Now the L ∞ -estimate, the hypothesis (3.3) and the properties of H allow to apply standardregularity results (see [23], [33]), in order to obtain that u ∈ C ,α (Ω). As matter of fact,as observed in [9] it is possible to follow the argument in [30, pages 466-467] to get the F. DELLA PIETRA, N. GAVITONE continuity of u p up to the boundary. Finally, u p is strictly positive in Ω by the Harnackinequality (see [34]). (cid:3) Theorem 3.2.
The first eigenvalue λ (Ω) of (3.1) is simple, that is the relative eigenfunc-tions are unique up to a multiplicative constant.Proof. We follow the idea of [4, 5]. Let v, w two positive minimizers of (1.1) in Ω such that k v k p = k w k p = 1, and consider η t = ( tv p + (1 − t ) w p ) /p , with t ∈ [0 , k η t k p = 1.Moreover, using the homogeneity and the convexity of H we get that(3.6) [ H ( Dη t )] p = η pt (cid:20) H (cid:18) t (cid:18) vη t (cid:19) p Dvv + (1 − t ) (cid:18) wη t (cid:19) p Dww (cid:19)(cid:21) p = η pt (cid:20) H (cid:18) s ( x ) Dvv + (1 − s ( x )) Dww (cid:19)(cid:21) p ≤ η pt (cid:20) s ( x ) H (cid:18) Dvv (cid:19) + (1 − s ( x )) H (cid:18) Dww (cid:19)(cid:21) p ≤ tv p (cid:20) H (cid:18) Dvv (cid:19)(cid:21) p + (1 − t ) w p (cid:20) H (cid:18) Dww (cid:19)(cid:21) p = t [ H ( Dv )] p + (1 − t )[ H ( Dw )] p . Hence, recalling (1.2), the inequalities in (3.6) and the definition of η t give that J ( η t ) ≤ tJ ( v ) + (1 − t ) J ( w ) = λ (Ω) , and then η t is a minimizer for J . This implies that the inequalities in (3.6) become equalities.The equality between the third and the fourth row of (3.6) holds if and only if H ( Dv/v ) = H ( Dw/w ). Hence, the strict convexity of the level sets of H guarantees from the equalitiesin (3.6) that Dv/v = Dw/w in Ω, that is v/w is constant. The norm constraint on v and w implies the uniqueness, and this concludes the proof. (cid:3) Remark 3.1.
We stress that the nonnegative solution u p ∈ C ,α (Ω) ∩ C ( ¯Ω) of (3.1) wefound by Theorem 3.1 cannot be identically zero on ∂ Ω. Indeed, in such a case, taking ψ = 1as test function in (3.2), we obtain Z Ω u p − p dx = 0 , contradicting the positivity of u p in Ω. As a matter of fact, if we suppose ∂ Ω to be a connected C manifold, then the Hopf boundary point Lemma holds (see [14]), which implies that u cannot vanish on ∂ Ω. Theorem 3.3.
Any nonnegative function v ∈ W ,p (Ω) , v , which satisfies, in the senseof (3.2) , (3.7) ( − div ( F p ( Dv )) = λv p − in Ω ,F p ( Dv ) · ν + βH ( ν ) v p − = 0 on ∂ Ω . is a first eigenfunction of (3.7) , that is λ = λ (Ω) and v = u p , where u p is given in Theorem3.1, up to multiplicative constant. For analogous results in the Dirichlet case, see for example [28] and the references therein.
Proof of Theorem 3.3.
The same arguments of Theorem 3.1 allow to prove that the givennonnegative solution v of (3.7) is in C ,α (Ω) ∩ C ( ¯Ω) and it is positive in Ω. Moreover, thefunction u p ∈ C ,α (Ω) ∩ C ( ¯Ω) satisfies(3.8) Z Ω [ H ( Du p )] p dx + β Z ∂ Ω u pp H ( ν ) dσ = λ (Ω) Z Ω u pp dx, ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 7 while, choosing u pp / ( v + ε ) p − , with ε >
0, as test function for v , we get(3.9) Z Ω p (cid:20) H (cid:16) u p v + ε Dv (cid:17)(cid:21) p − H ξ ( Dv ) · Du p dx − ( p − Z Ω (cid:20) H (cid:16) u p v + ε Dv (cid:17)(cid:21) p dx ++ β Z ∂ Ω v p − ( v + ε ) p − u pp H ( ν ) dσ = λ Z Ω v p − ( v + ε ) p − u pp dx. Subtracting (3.9) by (3.8), being H ξ zero homogeneous, and observing that v/ ( v + ε ) ≤ Z Ω (cid:26) [ H ( Du p )] p − F p (cid:16) u p v + ε Dv (cid:17) · Du p + ( p − (cid:20) H (cid:16) u p v + ε Dv (cid:17)(cid:21) p (cid:27) dx ≤≤ Z Ω (cid:20) λ (Ω) − v p − ( v + ε ) p − λ (cid:21) u pp dx. The convexity of H p guarantees that the left-hand side in the above inequality is nonnegative.Hence, as ε →
0, the monotone convergence gives that( λ (Ω) − λ ) Z Ω u pp dx ≥ , and this can hold if and only if λ ≤ λ (Ω). Being λ (Ω) the smallest possible eigenvalue,necessarily we have that λ = λ (Ω). The uniqueness of the first eigenfuction implies that,up to some positive multiplicative constant, v = u p . (cid:3) In order to show a lower bound for λ (Ω) when Ω is a convex set of R n in terms of theanisotropic inradius of Ω, we need an Hardy-type inequality for functions which, in general,do not vanish on the boundary. To this aim, we impose further regularity on H . Moreprecisely, we assume also that(3.10) ∂W = { x : H o ( x ) = 1 } has positive Gaussian curvature in any point.If Ω is C , this assumption ensures that the anisotropic distance from the boundary of Ω is C in a tubular neighborhood of ∂ Ω (see for instance [13]).
Lemma 3.1.
Let Ω be a bounded convex open set of R n with C boundary and suppose that H o satisfies also (3.10) . Then, for any α > and ϑ > , the following Hardy-type inequalityholds: (3.11) Z Ω [ H ( Du )] p dx + ϑ p − Z ∂ Ω | u | p H ( ν ) dσ ≥ ( p − αϑ ) p − (1 − αϑ ) Z Ω | u | p ( d H + α ) p dx, where u ∈ W ,p (Ω) and d H is the anisotropic distance from the boundary of Ω , defined in (2.9) .Proof. It sufficient to prove the thesis for u ≥
0. Moreover, using an approximation argument,we can suppose that u ∈ C ( ¯Ω). For δ positive, let us define H δ ( ξ ) = H δ ( ξ ) + δ , where H δ isthe δ -mollification of H . By the convexity of H ( ξ ), the function H δ is convex and we have,for any ξ , ξ ∈ R n ,[ H δ ( ξ )] p ≥ [ H δ ( ξ )] p + p [ H δ ( ξ )] p − ( H δ ) ξ ( ξ ) · ( ξ − ξ ) . We apply the above inequality to ξ = Du and ξ = αϑud ε + α Dd ǫ , where α > ϑ >
0, and d ǫ is the ǫ -mollification of d H . The convexity of Ω gives that the function d H , and then d ǫ , are F. DELLA PIETRA, N. GAVITONE concave functions. We have:(3.12) Z Ω [ H δ ( Du )] p dx ≥ ( αϑ ) p Z Ω u p ( d ǫ + α ) p [ H δ ( Dd ǫ )] p dx ++ p ( αϑ ) p − Z Ω u p − ( d ǫ + α ) p − [ H δ ( Dd ǫ )] p − ( H δ ) ξ ( Dd ǫ ) · Du dx + − p ( αϑ ) p Z Ω u p ( d ǫ + α ) p [ H δ ( Dd ǫ )] p − ( H δ ) ξ ( Dd ǫ ) · Dd ǫ dx Passing to the limit as δ → − ( p − αϑ ) p Z Ω u p ( d ǫ + α ) p [ H ( Dd ǫ )] p dx. Moreover, by the divergence theorem we have that(3.13) p Z Ω u p − ( d ǫ + α ) p − [ H δ ( Dd ǫ )] p − ( H δ ) ξ ( Dd ǫ ) · Du dx == 1 p Z Ω d ǫ + α ) p − ( H pδ ) ξ ( Dd ǫ ) · D ( u p ) dx == 1 p Z ∂ Ω u p ( d ǫ + α ) p − ( H pδ ) ξ ( Dd ǫ ) · ν dσ − p Z Ω u p div (cid:18) ( H pδ ) ξ ( Dd ǫ )( d ǫ + α ) p − (cid:19) dx == 1 p Z ∂ Ω u p ( d ǫ + α ) p − ( H pδ ) ξ ( Dd ǫ ) · ν dσ − p Z Ω u p ( d H + α ) p − div(( H pδ ) ξ ( Dd ǫ )) dx ++ p − p Z Ω u p ( d ǫ + α ) p ( H pδ ) ξ ( Dd ǫ ) · Dd ǫ dx ≥≥ p Z ∂ Ω u p ( d ǫ + α ) p − ( H pδ ) ξ ( Dd ǫ ) · ν dσ + p − p Z Ω u p ( d ǫ + α ) p ( H pδ ) ξ ( Dd ǫ ) · Dd ǫ dx Last inequality follows from the fact that − div(( H pδ ) ξ ( Dd ǫ )) is nonnegative. Indeed, it is thetrace of the product of the matrices (cid:2) ( H pδ ) ξξ ( Dd ǫ ) (cid:3) and (cid:2) − D d ǫ (cid:3) , which are both positivesemidefinite, being H pδ convex and d ǫ concave.Passing to the limit as δ → p Z Ω u p − ( d ǫ + α ) p − [ H ( Dd ǫ )] p − ( H ) ξ ( Dd ǫ ) · Du dx ≥≥ p Z ∂ Ω u p ( d ǫ + α ) p − ( H p ) ξ ( Dd ǫ ) · ν dσ + ( p − Z Ω u p ( d ǫ + α ) p [ H ( Dd ǫ )] p dx Then, as δ → Z Ω [ H ( Du )] p dx − ( αϑ ) p − p Z ∂ Ω u p ( H p ) ξ (cid:18) Dd ǫ d ǫ + α (cid:19) · ν dσ ≥≥ ( p − αϑ ) p − (1 − αϑ ) Z Ω u p ( d ǫ + α ) p [ H ( Dd ǫ )] p dx. Now we pass to the limit for ǫ →
0. Recalling that under our assumptions d H is C in atubular neighborhood of ∂ Ω, by uniform convergence we get(3.15) Z Ω [ H ( Du )] p dx − ( αϑ ) p − Z ∂ Ω u p ( d H + α ) p − [ H ( Dd H )] p − H ξ ( Dd H ) · νdσ ≥≥ ( p − αϑ ) p − (1 − αϑ ) Z Ω u p ( d H + α ) p [ H ( Dd H )] p dx. ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 9
Being H ( Dd H ) = 1 a.e. in Ω, and d H = 0 on ∂ Ω, choosing ϑ p − = β , and recalling that ν = − Dd H / | Dd H | on ∂ Ω, by (2.1) and (2.5) we get the thesis. (cid:3)
An immediate application of the previous Lemma is the following result.
Proposition 3.1. If Ω is a convex set of R n with C boundary and if H satisfies also (3.10) ,then λ (Ω) ≥ (cid:18) p − p (cid:19) p βR H, Ω (cid:16) β p − R H, Ω (cid:17) p − , where R H, Ω is the anisotropic inradius of Ω , as defined in (2.11) .Proof. Let β = ϑ p − . Then, by (3.11) and the definitions of λ (Ω) and of the anisotropicinradius R H, Ω we get that λ (Ω) ≥ ( p − β ( R H, Ω + α ) p (1 − β p − α ) α p − . Then, maximizing the right-hand side of the above inequality we obtain that λ (Ω) ≥ (cid:18) p − p (cid:19) p βR H, Ω (cid:16) β p − R H, Ω (cid:17) p − . (cid:3) Remark 3.2.
As a consequence of the previous Proposition, we have thatsup | Ω | = m λ (Ω) = + ∞ . among all the Lipschitz domains with given measure m > Proposition 3.2.
For any t > , we have that λ ( t Ω , β ) = t − p λ (Ω , t p − β ) .Proof. By the homogeneity of H , we have: λ ( t Ω , β ) = min v ∈ W ,p ( t Ω) v =0 Z t Ω [ H ( Dv ( x ))] p dx + β Z ∂ ( t Ω) | v ( x ) | p H ( ν ( x )) dσ ( x ) Z t Ω | v ( x ) | p dx = min u ∈ W ,p (Ω) u =0 t n − p Z Ω [ H ( Du ( y ))] p dy + t n − β Z ∂ Ω | u ( y ) | p H ( ν ( y )) dσ ( y ) t n Z Ω | u ( y ) | p dy = t − p λ (Ω , t p − β ) . (cid:3) The eigenvalue problem in the anisotropic radial case
In this section we study the properties of the minimizers of (1.1) when Ω is homotheticto the Wulff shape, that is, for
R >
0, the functions v p such that(4.1) J ( v p ) = min u ∈ W ,p ( W R ) u Z W R [ H ( Du )] p dx + β Z ∂ W R | u | p H ( ν ) dσ Z W R | u | p dx , where W R = R W = { x : H o ( x ) < R } , with R >
0, and W is the Wulff shape centered atthe origin. By Theorem 3.1, such functions solve the following problem:(4.2) ( − div ( F p ( Dv )) = λ ( W R ) | v | p − v in W R ,F p ( Dv ) · ν + βH ( ν ) | v | p − v = 0 on ∂ W R . Theorem 4.1.
Let v p ∈ C ,α (Ω) ∩ C ( ¯Ω) be a positive solution of problem (4.2) . Then, thereexists a decreasing function ̺ p = ̺ p ( r ) , r ∈ [0 , R ] , such that ̺ p ∈ C ∞ (0 , R ) ∩ C ([0 , R ]) , and v p ( x ) = ̺ p ( H o ( x )) , x ∈ W R ,̺ ′ p (0) = 0 , − ( − ̺ ′ p ( R )) p − + β ( ̺ p ( R )) p − = 0 . Proof.
Let B R be the Euclidean ball centered at the origin, B R = { x ∈ R n : | x | < R } , andconsider the p -Laplace eigenvalue problem in B R , that is (4.2) with H ( ξ ) = | ξ | :(4.3) ( − ∆ p w = λ , E ( B R ) | w | p − w in B R , | Dw | p − ∂w∂ν + β | w | p − w = 0 on ∂B R , where λ , E ( B R ) denotes the first eigenvalue. It is known (see, for example, [16]) that problem(4.3) admits a positive radially decreasing solution w p ( x ) = ̺ p ( | x | ), 0 ≤ | x | ≤ R , such that ̺ p ∈ C ∞ (0 , R ) ∩ C ([0 , R ]) and verifies(4.4) − ( p − − ̺ ′ p ( r )) p − ̺ ′′ p ( r ) + n − r ( − ̺ ′ p ( r )) p − = λ , E ( B R ) ̺ p ( r ) p − , r ∈ ]0 , R [ ,̺ ′ p (0) = 0 , − ( − ̺ ′ p ( R )) p − + β̺ p ( R ) p − = 0 . Let v p ( x ) = ̺ p ( H o ( x )), x ∈ W R . Using properties (2.5)–(2.7), for x ∈ W R \ { } we have that H ( Dv p ( x )) = − ̺ ′ p ( H o ( x )) H ( DH o ( x )) = − ̺ ′ p ( H o ( x )) , and DH ( Dv p ( x )) = − DH ( DH o ( x )) = − xH o ( x ) , which imply that(4.5) F p ( Dv p ) = − ( − ̺ ′ ( H o ( x ))) p − xH o ( x ) , and then, by (4.4), − div( F p ( Dv p )) = − ( p − − ̺ ′ p ( H o ( x ))) p − ̺ ′′ p ( H o ( x )) + n − H o ( x ) ( − ̺ ′ p ( H o ( x ))) p − = λ , E ( B R ) v p ( x ) p − for x ∈ W R \ { } . (4.6)As regards the boundary condition, observing that ν ( x ) = DH o ( x ) / | DH o ( x ) | , by (4.5), theproperties (2.5), (2.6), and (4.4) we have that F p ( v p ( x )) · ν ( x ) + βH ( ν ( x )) v p ( x ) p − = 1 | DH o ( x ) | (cid:16) − ( − ̺ ′ ( R )) p − + β̺ p ( R ) p − (cid:17) = 0 for x ∈ ∂ W R . (4.7)Hence, integrating (4.6) on W R \ W ε , we can use the divergence theorem and the boundarycondition (4.7), and let ε going to 0, obtaining that v p verifies(4.8) ( − div ( F p ( Dv p )) = λ , E ( B R ) v p − p in W R ,F p ( Dv ) · ν + βH ( ν ) v p − p = 0 on ∂ W R . ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 11
But Theorem 3.3 guarantees that a positive solution of (4.8) has to be a first eigenfunction,and λ ( W R ) = λ , E ( B R ) . This concludes the proof. (cid:3)
Remark 4.1.
We observe that the proof of the above theorem shows that, for any convexfunction H we can consider, the first eigenvalue in the ball W R = { H o ( x ) < R } is the same,and coincides with the first eigenvalue for the p -Laplacian problem (4.3) in the Euclideanball B R (with the same R ).Next two lemmata will be useful in the proof of the main result. Their proofs are analogousto the ones obtained in [9]. For the sake of completeness, we write them in details. Lemma 4.1. If < r < s , then λ ( W r ) > λ ( W s ) .Proof. Let v p a minimizer of (4.1), with R = r , and take w ( x ) = v p (cid:0) rs x (cid:1) , x ∈ W s . Then, bythe homogeneity of H we get λ ( W s ) ≤ Z W s [ H ( Dw )] p dx + β Z ∂ W s | w | p H ( ν ) dσ Z W s | w | p dx = (cid:16) rs (cid:17) p Z W r [ H ( Dv p )] p dx + β rs Z ∂ W r | v p | p H ( ν ) dσ Z W r | v p | p dx< Z W r [ H ( Dv p )] p dx + β Z ∂ W r | v p | p H ( ν ) dσ Z W r | v p | p dx = λ ( W r ) (cid:3) We stress that by (4.4), if v p ( x ) = ̺ p ( H o ( x )) is the positive solution in W R we found inTheorem 4.1, we have that, for x ∈ ∂ W R , β = (cid:2) H (cid:0) Dv p ( x ) (cid:1)(cid:3) p − v p ( x ) p − . Then, for every 0 ≤ r ≤ R , we define(4.9) β r = (cid:2) H (cid:0) Dv p ( x ) (cid:1)(cid:3) p − v p ( x ) p − , for H o ( x ) = r. Let us observe that β = 0 and β R = β . Lemma 4.2. If ≤ r < s ≤ R , then β r < β s .Proof. We first observe that, similarly as in the proof of Theorem 4.1, for 0 < r < R , thefunction v p is such that ( − div ( F p ( Dv p )) = λ ( W R ) v p − p in W r ,F p ( Dv p ) · ν + β r H ( ν ) v p − p = 0 on ∂ W r . Then, denoted by λ ( W r , β r ) the first eigenvalue in W r with β = β r , by Theorem 3.3 wehave necessarily λ ( W R ) = λ ( W r , β r ) for all r ∈ ]0 , R ]. Hence, by Lemma 4.1 we obtain, for0 < r < s ≤ R , that Z W r [ H ( Dv p )] p dx + β r Z ∂ W r v pp H ( ν ) dσ Z W r v pp dx = λ ( W r , β r ) = λ ( W s , β s ) < λ ( W r , β s ) ≤≤ Z W r [ H ( Dv p )] p dx + β s Z ∂ W r v pp H ( ν ) dσ Z W r v pp dx , and then β r < β s . (cid:3) A representation formula for λ (Ω)Now we prove a level set representation formula for the first eigenvalue λ (Ω). To thisaim, we will use the following notation. Let ˜ u p be the first positive eigenfunction such thatmax ˜ u p = 1. Then, for t ∈ [0 , U t = { x ∈ Ω : ˜ u p > t } ,S t = { x ∈ Ω : ˜ u p = t } , Γ t = { x ∈ ∂ Ω : ˜ u p > t } . First of all, it is worth to observe that the anisotropic areas of the sets ∂U t , S t and Γ t , definedin 2.1, are related in the following way. Lemma 5.1.
There exists a countable set
Q ⊂ ]0 , such that (5.1) σ H ( ∂U t ) ≤ σ H (Γ t ) + σ H ( S t ) , ∀ t ∈ ]0 , \Q . Proof.
The proof follows similarly as in [9]. The continuity up to the boundary of theeigenfunction ˜ u p , given in Theorem 3.1, guarantees that ∂U t ∩ Ω ⊆ S t , ∂U t ∩ ∂ Ω ⊆ ˜Γ t for any t ∈ [0 , t = { x ∈ ∂ Ω : ˜ u p ≥ t } . Moreover, by [32, Section 1.2.3] we havethat Z ∞ σ H (Γ t ) dt = Z ∞ σ H (˜Γ t ) dt = Z ∂ Ω ˜ u dσ H ≤ σ H ( ∂ Ω) < + ∞ . Hence σ H (Γ t ) ≤ σ H (˜Γ t ) < + ∞ and then σ H (Γ t ) = σ H (˜Γ t ) for a.e. t ∈ [0 , σ H (Γ t ) and σ H (˜Γ t ) monotone decreasing in t , they are continuous in [0 ,
1] up to acountable set Q . Hence, σ H ( ∂U t ) = σ H ( ∂U t ∩ Ω) + σ H ( ∂U t ∩ ∂ Ω) ≤ σ H ( S t ) + σ H (Γ t ) , for all t ∈ [0 , \ Q . (cid:3) ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 13
If we formally divide both terms in the equation in (3.1) by ˜ u p − p , and integrate in U t , by(2.5) and the boundary condition we get(5.2) λ (Ω) | U t | = Z U t − div (cid:0) F p ( D ˜ u p ) (cid:1) ˜ u p − p dx == − ( p − Z U t [ H ( D ˜ u p )] p − H ξ ( D ˜ u p ) · D ˜ u p ˜ u pp dx − Z ∂U t [ H ( D ˜ u p )] p − ˜ u p − p H ξ ( D ˜ u p ) · ν dσ == − ( p − Z U t [ H ( D ˜ u p )] p ˜ u pp dx + Z S t [ H ( D ˜ u p )] p − ˜ u p − p H ( ν ) dσ + β Z Γ t H ( ν ) dσ == | U t |F Ω U t , [ H ( D ˜ u p )] p − ˜ u p − p ! , where(5.3) F Ω ( U t , ϕ ) = 1 | U t | (cid:18) − ( p − Z U t ϕ p ′ dx + Z S t ϕH ( ν ) dσ + β Z Γ t H ( ν ) dσ (cid:19) , with ϕ nonnegative measurable function in Ω. The formal computations in (5.2) give arepresentation formula of λ (Ω) which will be rigorously proved in the result below. Theorem 5.1.
Let ˜ u p ∈ C ,α (Ω) ∩ C ( ¯Ω) be the positive minimizer of (1.1) such that max ˜ u p = 1 . Then, for a.e. t ∈ ]0 , , (5.4) λ (Ω) = F Ω U t , [ H ( D ˜ u p )] p − ˜ u p − p ! . Proof.
Let 0 < ε < t <
1, and ψ ε = u p ≤ tu − tε u p − p if t < ˜ u p < t + ε u p − p if ˜ u p ≥ t + ε. The functions ψ ε are in W ,p (Ω) and increasingly converge to ˜ u − ( p − p χ U t as ε ց
0. Moreover, Dψ ε = u p < t ε (cid:18) ( p − t ˜ u p + 2 − p (cid:19) D ˜ u p ˜ u p − p if t < u p < t + ε − ( p − D ˜ u p ˜ u pp if ˜ u p > t + ε. Then, choosing ψ ε as test function in (3.2), we get that the first integral is − ( p − Z U t + ε [ H ( D ˜ u p )] p ˜ u pp dx + 1 ε Z U t \ U t + ε [ H ( D ˜ u p )] p ˜ u p − p (cid:18) ( p − t ˜ u p + 2 − p (cid:19) dx == − ( p − Z U t + ε [ H ( D ˜ u p )] p ˜ u pp dx + 1 ε Z t + εt (cid:18) ( p − tτ + 2 − p (cid:19) Z S τ [ H ( D ˜ u p )] p − ˜ u p − p H ( ν ) dσ, where last equality follows by the coarea formula. Then, reasoning similarly as in [9], we getthat Z Ω [ H ( D ˜ u p )] p − H ξ ( D ˜ u p ) · Dψ ε dx ε → −−−→ − ( p − Z U t [ H ( D ˜ u p )] p ˜ u pp dx + Z S t [ H ( D ˜ u p )] p − ˜ u p − p H ( ν ) dσ. As regards the other two integrals in (3.2), we have that β Z ∂ Ω ˜ u p − p ψ ε H ( ν ) dσ = β Z Γ t + ε H ( ν ) dσ + β Z Γ t \ Γ t + ε u − tε H ( ν ) dσ ε → −−−→ β Z Γ t H ( ν ) dσ, and, by monotone convergence theorem and the definition of ψ ε , λ (Ω) Z Ω ˜ u p − p ψ ε dx ε → −−−→ λ (Ω) | U t | . Summing the three limits, we get (5.4). (cid:3)
Theorem 5.2.
Let ϕ be a nonnegative function in Ω such that ϕ ∈ L p ′ (Ω) . If ϕ [ H ( D ˜ u p )] p − / ˜ u p − p , where ˜ u p is the eigenfunction given in Theorem 5.1, and F Ω is the func-tional defined in (5.3) , then there exists a set S ⊂ ]0 , with positive measure such that forevery t ∈ S it holds that (5.5) λ (Ω) > F Ω ( U t , ϕ ) . Proof.
The proof is similar to the one obtained in [9], and we only sketch it here. It can bedivided in two main steps. First, we claim that, if w ( x ) := ϕ − [ H ( D ˜ u p )] p − ˜ u p − p , I ( t ) := Z U t w H ( D ˜ u p )˜ u p dx, then I : ]0 , → R is locally absolutely continuous and(5.6) F Ω ( U t , ϕ ) ≤ λ (Ω) − | U t | t p − (cid:16) ddt t p I ( t ) (cid:17) , for almost every t ∈ ]0 , ,
1[ of positive measure.In order to prove (5.6), writing the representation formula (5.4) in terms of w , it followsthat, for a.e. t ∈ ]0 , F Ω ( U t , ϕ ) = λ (Ω) + 1 | U t | (cid:18)Z S t wH ( ν ) dσ − ( p − Z U t (cid:16) ϕ p ′ − [ H ( D ˜ u p )] p ˜ u pp (cid:17) dx (cid:19) ≤ λ (Ω) + 1 | U t | (cid:18)Z S t wH ( ν ) dσ − p Z U t w H ( D ˜ u p )˜ u p dx (cid:19) = λ (Ω) + 1 | U t | (cid:18)Z S t wH ( ν ) dσ − p I ( t ) (cid:19) (5.7)where the inequality in (5.7) follows from the inequality ϕ p ′ ≥ v p ′ + p ′ v p ′ − ( ϕ − v ), with ϕ, v ≥
0. Applying the coarea formula, it is possible to rewrite I ( t ) as I ( t ) = Z U t w H ( D ˜ u p )˜ u p dx = Z t τ dτ Z S τ w H ( ν ) dσ. This assures that I ( t ) is locally absolutely continuous in ]0 ,
1[ and, for almost every t ∈ ]0 , − ddt (cid:0) t p I ( t ) (cid:1) = t p − (cid:18)Z S t w H ( ν ) dσ − pI ( t ) (cid:19) . Substituting in (5.7), the inequality (5.6) follows. In order to conclude the proof, arguing bycontradiction exactly as in [9, Theorem 3.2], it is possible to show that G ( t ) := t p I ( t ) haspositive derivative in a set of positive measure. Together with (5.6), this implies (5.5). (cid:3) ABER-KRAHN INEQUALITY FOR ANISOTROPIC ROBIN PROBLEMS 15 Main result
Now we are in position to state and prove the desired Faber-Krahn inequality.
Theorem 6.1.
Let Ω ⊂ R n , n ≥ , be a bounded Lipschitz domain, and H : R n → [0 , + ∞ [ a function with strictly convex sublevel sets which satisfies (2.1) , (2.2) , and (3.3) . Then, (6.1) λ (Ω) ≥ λ ( W R ) , where W R is the Wulff shape centered at the origin such that |W R | = | Ω | . The equality holdsif and only if Ω is a Wulff shape.Proof. The first step in order to prove the result is to construct a suitable test function in Ωfor (5.3). Let v p be a positive eigenfunction of the anisotropic radial problem (4.2) in W R .By Theorem 4.1, v p is a function depending only by H o ( x ), and then we are able to define,as in (4.9), the function β r = ϕ ⋆ ( x ) = [ H ( Dv p ( x ))] p − v p ( x ) p − , with x ∈ W R , i.e. H o ( x ) = r ∈ [0 , R ] . As before, let ˜ u p be the first eigenfunction of (3.1) in Ω such that k ˜ u p k ∞ = 1. Using thesame notation of Section 5, for any t ∈ ]0 ,
1[ we consider W r ( t ) , the Wulff shape centered atthe origin, where r ( t ) is the positive number such that | U t | = |W r ( t ) | . Then, for x ∈ Ω and˜ u p ( x ) = t , we define ϕ ( x ) := β r ( t ) . Similarly as in [9], ϕ is a measurable function. Thanks to this test function, we can compare F Ω ( U t , ϕ ) with F W R ( B r ( t ) , ϕ ⋆ ). Indeed, we claim that F Ω ( U t , ϕ ) ≥ |W r ( t ) | − ( p − Z W r ( t ) ϕ p ′ ⋆ dx + Z ∂ W r ϕ ⋆ H ( ν ) dσ ! = F W R ( W r ( t ) , ϕ ⋆ )(6.2)for all t ∈ ]0 , \Q , where Q is the set of Lemma 5.1. In order to show (6.2), we first observethat by [32, Section 1.2.3], being | U t | = |W r ( t ) | for all t ∈ ]0 , Z U t ϕ p ′ dx = Z W r ( t ) ϕ p ′ ⋆ dx. Moreover, the anisotropic isoperimetric inequality (2.8), Lemma 5.1 and being, by Lemma4.2, β r ( t ) ≤ β for any t , we have that(6.4) Z ∂ W r ( t ) ϕ ⋆ H ( ν ) dσ = β r ( t ) σ H ( ∂ W r ( t ) ) ≤≤ β r ( t ) σ H ( ∂U t ) ≤ β r ( t ) σ H ( S t ) + β r ( t ) σ H (Γ t ) ≤ Z S t ϕH ( ν ) dσ + β Z Γ t H ( ν ) dσ. Hence, joining (6.3) and (6.4) we get (6.2). Then, applying the level set representationformula (5.4) in the anisotropic radial case, and (5.5), by (6.2) we get λ ( W R ) = F W R ( W r ( t ) , ϕ ⋆ ) ≤ F Ω ( U t , ϕ ) ≤ λ (Ω)for some t ∈ ]0 , λ (Ω) = λ ( W R ).We first claim that, for a.e. t ∈ ]0 , U t is homothetic to a Wulff shape. Indeed, by (5.5)and (6.2) λ ( W R ) = λ (Ω) ≥ F Ω ( U t , ϕ ) ≥ F W R ( W r ( t ) , ϕ ⋆ ) = λ ( W R ) for t in a set of positive measure S ⊂ ]0 , ϕ = H ( D ˜ u p ) p − ˜ u p − p . This implies that for almost every t ∈ ]0 , σ H ( ∂ W r ( t ) ) = σ H ( ∂U t ) for a.e. t . By the equality case in theanisotropic isoperimetric inequality, we get the claim. Since U t , t ∈ ]0 ,
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Francesco Della Pietra, Universit`a degli studi di Napoli “Federico II”, Dipartimento di Ma-tematica e Applicazioni “R. Caccioppoli”, Complesso di Monte Sant’Angelo, Via Cintia, 80126Napoli, Italia.
E-mail address : [email protected] Nunzia Gavitone, Universit`a degli studi di Napoli “Federico II”, Dipartimento di Matemati-ca e Applicazioni “R. Caccioppoli”, Complesso di Monte Sant’Angelo, Via Cintia, 80126 Napoli,Italia.
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