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Bounds for the rate of change of the Sobolev constant

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If you have a question about this talk, please contact Andrew Morris.

The Sobolev constant of a bounded region in Euclidean space decreases as the region expands. Isoperimetric bounds for the rate of decrease of the Sobolev constant are obtained. These apply, in particular, to the first Dirichlet eigenfunction of the Laplacian and to the (inverse of the) torsional rigdity. The main ingredients are a fairly standard Hadamard variation formula and a reverse Holder inequality for the eigenfuntions. This is joint work with Jesse Ratzkin and Moustapha Fall.

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