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arXiv · 2504.09637

Optimal convergence rates for the finite element approximation of the Sobolev constant

Liviu I. Ignat·Enrique Zuazua
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Abstract

We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the Sobolev deficit in suitable quasi-norms, which have been introduced and utilized in the context of finite element approximations of the p-Laplacian. The proof further involves sharp estimates for the finite element approximation of Sobolev minimizers.

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Liviu I. Ignat, Enrique Zuazua. 2025-04-13. Optimal convergence rates for the finite element approximation of the Sobolev constant. https://arxiv.org/abs/2504.09637

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