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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
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