## Rearrangement inequalities and applications to isoperimetric problems for eigenvalues.(English)Zbl 1234.35161

The aim of this paper is to present a new rearrangement technique and to shown how this technique can be used to obtain new results of Faber-Krahn type for symmetric and nonsymmetric second-order operators. Let $$\Omega \subset R^n$$ be a sufficiently smoothly bounded domain, the authors consider the operator $L = - \text{div}(A\nabla)+v\nabla+V,$ where $$A$$ is a symmetric matrix field in $$W^{1,\infty}(\Omega)$$, $$V$$ is a continuous function in $$\overline{\Omega}$$ and $$v$$ is a vector field in $$L^\infty(\Omega,R^n)$$.
For a given $$\Omega$$, $$A$$, $$v$$, $$V$$ such that $$A \geq \Lambda (x) \text{ Id}$$ on $$\Omega$$ in the sense of quadratic forms, Id denotes the $$n \times n$$ identity matrix, and the principal eigenfunction $$\varphi$$ of the operator $$L$$ solves $\begin{cases} -\text{div}(A\nabla \varphi ) + v\nabla\varphi + V \varphi = \lambda_1 \varphi &\text{in } \Omega, \\ \varphi > 0 &\text{in } \Omega, \\ \varphi = 0 &\text{on } \partial \Omega, \\ \|\varphi\|_{L^{\infty}(\Omega)} = 1,\end{cases}$ we associate the rearranged functions or vector fields $$\tilde{\varphi}$$, $$\widehat{\Lambda}$$, $$\widehat{v}$$, $$\widehat{V}$$ defined on $$\Omega^*$$, where $$\Omega^*$$ denotes the Euclidean ball centered at $$0$$ and having the same Lebesgue measure as $$\Omega$$. Roughly speaking the main ideas of this new rearrangement technique work as follow. For $\Omega_a = \{x \in \Omega, a < \varphi(x) \leq 1 \}, \qquad 0 \leq a < 1,$ and $$\rho (a) \in (0,R]$$, such that $$|\Omega_a| = |B_{\rho(a)}|$$ ($$B_s$$ denotes the open Euclidean ball of radius $$s>0$$ centered at $$0$$, $$R$$ denotes the radius of $$\Omega^*$$), the function $$\rho$$ is decreasing, continuous, one-to-one and onto and the rearrangement $$\tilde{\varphi}$$ of $$\varphi$$ satisfies $\int_{\Omega_a} \text{div}(A\nabla \varphi)(x) dx = \int_{B_{\rho(a)}} \text{div}(\widehat{\Lambda}\nabla \tilde{\varphi})(x)dx, \;0 \leq a < 1.$ For $$\tilde{\varphi}$$ it follows that, for all $$x \in \overline{\Omega^*}$$, $\tilde{\varphi}(x) \geq \rho^{-1}(|x|).$ This new rearrangement is in general different from the Schwarz symmetrization. The paper contains results comparing $$\lambda_1(\Omega,A,v,V)$$ with $$\lambda_1$$ for $$\Omega^*$$ and rearranged functions under different types of constrains, for example integral type and $$L^\infty$$ type.

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs

### Keywords:

Faber-Krahn inequality; rearrangement
Full Text:

### References:

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