A Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition

TL;DR

This paper establishes a Faber-Krahn inequality for the Laplacian with drift under Robin boundary conditions, demonstrating how eigenvalues behave as boundary parameters change and identifying optimal drifts under constraints.

Contribution

It introduces a Faber-Krahn inequality for the Laplacian with drift under Robin boundary conditions and characterizes optimal drifts minimizing or maximizing eigenvalues.

Findings

Faber-Krahn inequality holds for Robin boundary conditions with large enough beta.

Eigenvalues converge to Dirichlet eigenvalues as beta approaches infinity.

Existence and uniqueness of drifts minimizing or maximizing eigenvalues under constraints.

Abstract

We prove a Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition, provided that the $\beta$ parameter in the Robin condition is large enough. The proof relies on a compactness argument, on the convergence of Robin eigenvalues to Dirichlet eigenvalues when $\beta$ goes to infinity, and on a strict Faber-Krahn inequality under Dirichlet boundary condition. We also show the existence and uniqueness of drifts $v$ satisfying some $L^\infty$ constraints and minimizing or maximizing the principal eigenvalue of $-\Delta+v\cdot\nabla$ in a fixed domain and with a fixed parameter $\beta>0$ in the Robin condition.