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A Faber-Krahn inequality for Robin problems in any space dimension. (English) Zbl 1220.35103
Summary: We prove a Faber-Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that amongst all Lipschitz domains of fixed volume, the ball has the smallest first eigenvalue. We prove the result in all space dimensions using ideas from [M.-H. Bossel, C. R. Acad. Sci., Paris, Sér. I Math. 302, 47–50 (1986; Zbl 0606.73018)], where a proof for smooth domains in the plane was given. The method does not involve the use of symmetrisation arguments. The results also imply variants of the Cheeger inequality for the first eigenvalue.

MSC:
35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
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