Existence of optimizers in a Sobolev inequality for vector fields

Rupert L. Frank; Michael Loss

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arXiv · 2107.06450

Existence of optimizers in a Sobolev inequality for vector fields

Rupert L. Frank·Michael Loss

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Abstract

We consider the minimization problem corresponding to a Sobolev inequality for vector fields and show that minimizing sequences are relatively compact up to the symmetries of the problem. In particular, there is a minimizer. An ingredient in our proof is a version of the Rellich--Kondrachov compactness theorem for sequences satisfying a nonlinear constraint.

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Rupert L. Frank, Michael Loss. 2021-07-14. Existence of optimizers in a Sobolev inequality for vector fields. https://arxiv.org/abs/2107.06450

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