Optimal Szeg\"o-Weinberger type inequalities

Denote with $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu e^{h\left(|x|\right)}u&\text{in}&\Omega&&\frac{\partial u}{\partial \nu}=0&\text{on}&\partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded and Lipschitz domain in $\mathbb{R}^{N}$. Under suitable assumption on $h$ we prove that the ball centered at the origin is the unique set maximizing $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ among all Lipschitz bounded domains $\Omega $ of $\mathbb{R}^{N}$ of prescribed $e^{h\left(|x|\right)}dx$-measure and symmetric about the origin. Moreover, an example in the model case $h\left(|x|\right) =|x|^{2},$ shows that, in general, the assumption on the symmetry of the domain cannot be dropped. In the one-dimensional case, i.e. when $\Omega$ reduces to an interval $(a,b),$ we consider a wide class of weights (including both Gaussian and anti-Gaussian). We then describe the behavior of the eigenvalue as the interval $(a,b)$ slides along the $x$-axis keeping fixed its weighted length.


Introduction
In [22] Kornhauser and Stakgold made a famous conjecture: among all planar simply connected domains, with fixed Lebesgue measure the first nontrivial eigenvalue of the Neumann Laplacian achieves its maximum value if and only if Ω is a disk. This conjecture was proved by Szegö in [27]. In [29] Weinberger generalized this result to any bounded smooth domain of R N . Adapting Weinberger arguments, similar inequalities for spaces of constant sectional curvature have been derived (see, e.g., [2] and [11]). Further, it is proved in [23] that the first nonzero Neumann eigenvalue is maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area.
In this paper we derive some sharp Szegö-Weinberger type inequalities for the first nontrivial eigenvalue, µ 1 (Ω; e h(|x|) ), of the following class of problems Here and in the sequel, Ω will denote a bounded domain in R N with Lipschitz boundary and ν the outward normal to ∂Ω. Since the degeneracy of the operator is given in terms of the radial function e h(|x|) , it appears natural to let Ω vary in the class of sets having prescribed γ h -measure, where Recently it has been proved in [13], that among all Lipschitz domains Ω in R N , which are symmetric about the origin and have fixed Gaussian measure, µ 1 (Ω; e −|x| 2 /2 ) achieves its maximum value if and only if Ω is the Euclidean ball. In the same paper it has been shown that µ 1 ((a, b); e −t 2 /2 ) is minimal when the interval (a, b) reduces to a half-line and maximal when it is centered at the origin, and it is strictly monotone as (a, b) slides between these extreme positions.
The first part of the present paper deals with a class of weighted eigenvalue problems in the form where a, b ∈ R with a < b and (1.4) q ∈ C 2 (R) , q (x) > 0 ∀x ∈ R.
We analyze the behavior of µ 1 ((a, b) ; q) as the interval (a, b) slides along the x-axis keeping fixed its q−length = b a q(x)dx . We prove that if q(x) = q(|x|) and it is decreasing on R + then µ 1 ((a, b) ; q) behaves like µ 1 ((a, b); e −t 2 /2 ), while if q(x) is increasing on R + then µ 1 ((a, b) ; q) behaves in the opposite way. Note that the method used here is different and more general than the one of [13]. For the precise statement see Theorem 1.1 below. It treats the Gaussian and anti-Gaussiantype weights in a unified way. We emphasize that, in contrast to the case N ≥ 2, no concavity assumptions are imposed on the weight function. This explains the different notation in (1.1) and (1.3).
We consider the function Denote by µ 1 (a) the eigenvalue µ 1 ((a, b (a)) ; q). Then we have the following Theorem 1.1. Assume that q satisfies (1.4) and is even. Then Now let us turn our attention to the N -dimensional problem (1.1). We assume that h fulfills the following set of hypotheses Our second main result is the following.
Theorem 1.2. Assume that assumptions (1.5) are in force. Then the ball centered at the origin is the unique set maximizing µ 1 (Ω; e h(|x|) ) among all Lipschitz bounded domains Ω of R N of prescribed γ h -measure and symmetric about the origin. Moreover, if the assumption on the symmetry of the domain is dropped, then, in general, the thesis does not hold true.
The paper is organized as follows. Section 2 contains some results from the theory of weighted rearrangements along with the definition of the suitable Sobolev spaces naturally associated to problem (1.1). Section 3 is devoted to Theorem 1.1. By a suitable change of variable, we firstly show that µ 1 ((a, b) ; q) coincides with λ 1 (a, b) ; q −1 , the first Dirichlet eigenvalue of problem (1.3) with respect the weight q −1 . In turn, we observe that the following problems and P m : have the same first eigenvalue, where y =: F (x) := x 0 q (t) dt, α := F (a) , β := F (b) and m is defined in (3.9). The advantages of studying P m in place of P q are twofold. First, as long as the interval (a, b) moves along the x−axis with fixed q−length, then (α, β) slides along the y− axis keeping fixed its Lebesgue measure. Second, the new equation contains a weight only in the zero order term, so it nicely behaves under reflection with respect to (α + β)/2. These two circumstances allow to evaluate the sign of the shape derivative of the first eigenvalue of P m and hence of µ 1 ((a, b) ; q). In Section 4 we prove Theorem 1.2. To this aim, we first study problem (1.1) in the radial case, i.e., when Ω = B R , the ball centered at the origin of radius R. We deduce that µ 1 (B R ; e h(|x|) ) is an eigenvalue of multiplicity N , and a corresponding set of linearly independent eigenfunctions is {w (|x|) x i |x| , i = 1, ..., N }, with an appropriate function w.
Following Weinberger's original idea, we then define G(r) = w(r) for 0 ≤ r ≤ R and G(r) = w(R) for r > R. Since the functions G (|x|) x i |x| , (i = 1, ..., N ), have mean value zero, we may use them in the variational characterization of µ 1 (Ω; e h(|x|) ). Then the result is achieved by symmetrization arguments.

Notation and preliminary results
Now we recall a few definitions and properties about weighted rearrangement. For exhaustive treatment on this subject we refer, e.g., to , [14], [20] and [26].
Let u : x ∈ Ω → R be a measurable function. We denote by m(t) the distribution function of while the decreasing rearrangement and the increasing rearrangement of u are defined respectively by where B r denotes the ball of center the origin and radius r , Ω ⋆ is the ball B r ⋆ such that γ h Ω ⋆ = γ h (B r ⋆ ) . By its very definition u ⋆ is a radial and radially decreasing function. Since u and u ⋆ are equimeasurable, Cavalieri's principle ensures We will also make use of the Hardy-Littlewood inequality, which states that The natural functional space associated to problem (1.1) is the weighted Sobolev space defined as follows Since Ω is a bounded domain, assumptions (1.5) ensures that for some c 1 , c 2 in (0, +∞). Therefore, by the classical embedding theorems on the customary Sobolev space H 1 (Ω), we have that H 1 (Ω; γ h ) is compactly embedded in L 2 (Ω; γ h ). Hence, by standard theory on self-adjoint compact operator, µ 1 (Ω; e h(|x|) ) admits the following variational characterization : On the other hand as Hilbert spaces they clearly do not coincide.

The one-dimensional case
Throughout this Section we will assume that condition (1.4) is fulfilled and that −∞ < a < b < +∞.
We consider the weighted Neumann eigenvalue problem We denote by Here we are interested in studying the behavior of µ 1 ((a, b) ; q) when the interval (a, b) slides along the x-axis, keeping fixed its weighted length b a q (x) dx. To this end consider an eigenfunction u 1 of (3.1) corresponding to µ 1 ((a, b) ; q) . We consider the weighted Dirichlet eigenvalue problem and we denote its first eigenvalue by λ 1 (a, b) ; q −1 .

Proof.
By differentiating the equation Integrating this from We observe that u = 0, so that u is an eigenfunction for (3.1) and this completes the proof. It is straightforward to verify that v 1 is a solution of (3.3) with λ = λ 1 if and only if w(y) := v 1 (x), is a solution of (3.8) Without loss of generality we may assume w > 0.

Proof. Set
So, if we set W := − w − w, we find . Let k 1 be the first eigenvalue for the problem Since α, α+β 2 (α, β) , we have k 1 > λ 1 . From the maximum principle we obtain that the solution W of the boundary value problem (3.12), with g given, is unique. Moreover, W ≡ 0 for g ≡ 0, W > 0 for g ≥ 0, g = 0 and W < 0 for g ≤ 0, g = 0. Hence, using the strong maximum principle, we get if m ′ (y) ≥ 0 (m ′ (y) ≤ 0) for y > 0, and g ≡ 0 is possible in either case if m =cost. on (α, β). This completes the proof of Lemma.

Lemma 3.3. It holds that
where l is defined in (3.15).
Proof of Theorem 1.1. Using (3.9) and condition b a u 1 qdx = 0, we have that q ′ (x) ≥ 0 ∀x ≥ 0 if and only if m ′ (y) ≤ 0 for y ∈ (0, c/2) . Further, using the above notation and Lemma 3.2 we have µ 1 (a) = λ 1 (a), now the assertion follows from Lemma 3.2 and Lemma 3.4. Lemma 3.2 also allows to obtain qualitative properties about the eigenfunction u 1 of problem (3.1).
Lemma 3.5. Let q be even and u 1 an eigenfunction to problem (3.1) with eigenvalue λ 1 and a + b > 0.
Now the assertions follow from Lemma 3.2, Lemma 3.4 and Theorem 1.1.

The N −dimensional case
Let us consider the problem (1.1) in B R , the ball centered at the origin with radius R, i.e. (4.1) The equation in (4.1) can be rewritten, using polar coordinates, as As well known, see, e.g., [24] and [12], the last equality is fulfilled if and only if Multiplying the left hand side of equation (4.3) by f r 2 , we get Let us denote by f k , Y k the solutions of (4.3) with k = k defined in (4.4). The eigenfunctions are either purely radial or have the form The functions f k , with k ∈ N ∪ {0} , clearly satisfy (4.7) In the sequel we will denote by τ n (R), with n ∈ N∪{0}, the increasing sequence of eigenvalues of (4.1 ) whose corresponding eigenfunctions are purely radial, i.e. in the form ( 4.5) or equivalently solutions to problem (4.7) with k = 0. Clearly in this case the first eigenfunction is constant and the corresponding eigenvalue τ 0 (R) is trivially zero. We will denote by ν n (R), with n ∈ N, the remaining eigenvalues of (4.1), arranged in increasing order.
On the other hand, the first nontrivial eigenfunction of problem (4.9), g 1 = g 1 (r), has mean value zero i.e.
where, here and in the sequel, ω N denotes the Lebesgue measure of the unit ball in R N .

Now we claim that
To this aim we first note that Subtracting the above equalities and using The Lemma is so proved since We define where w is the solution of (4.9) satisfying (4.15). By the results stated above the function G is nondecreasing and nonnegative. We introduce the functions The assumption on the symmetry of Ω guarantees Hence each function P i is admissible in the variational formulation of µ 1 (Ω; e h(|x|) ), i.e. (2.3). Since where δ ij is the Kronecker symbol. Using P i as trial functions for µ 1 (Ω; e h(|x|) ) we get Taking into account of the definition (4.16) of G, and the differential equation in (4.10), with ν = ν 1 , we have Now we claim that (4.19) Hardy-Littlewood inequality (2.1) ensures (4.20) where N * is the decreasing rearrangement of N .
Note that N * (γ h (B r )) = N (r), since N * (γ h (B r )) and N (r) are equimeasurable and both radially decreasing functions. Therefore (4.21) N ω N Combining (4.20) and (4.21), we obtain the claim (4.19). Analogously it is possible to prove that Indeed since D is an increasing function, we have where D * is the increasing rearrangement of D. By (4.16 ), (4.19) and (4.22), (4.18) becomes which is the desired inequality. Moreover, from the monotonicity properties of the functions N and D, it easy to realize that inequalities (4.19) and (4.22) reduce to equalities only when Ω is the ball B R . We finally exhibit an example showing that, in general, the condition about the symmetry of the domain cannot be dropped. Let H n (t) := (−1) n e t 2 d n dt n e −t 2 , with t ∈ R, and v n (t) := H n (t)e −t 2 .
As recalled in Section 2, µ 1 (B r T , γ 2 ) satisfies the following variational characterization (4.25) µ 1 (B r T ; γ 2 ) = min In order to get an estimate from above for µ 1 (B r T ; γ 2 ) we use v = x and v = y as trial functions in (4.25) obtaining Summing up we get where the last inequality can be obtained by an elementary computation, taking into account of (4.23).
Hence the claim is proved.
Remark 4.1. Note that the assumption on the symmetry of Ω is used solely to guarantee the orthogonality conditions (4.17).