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arXiv · 2408.03299

Convergence rates of the fractional to the local Dirichlet problem

Leon Bungert·Félix del Teso
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Abstract

We prove non-asymptotic rates of convergence in the $W^{s,2}(\mathbb R^d)$-norm for the solution of the fractional Dirichlet problem to the solution of the local Dirichlet problem as $s\uparrow 1$. For regular enough boundary values we get a rate of order $\sqrt{1-s}$, while for less regular data the rate is of order $\sqrt{(1-s)|\log(1-s)|}$. We also obtain results when the right hand side depends on $s$, and our error estimates are true for all $s\in(0,1)$. The proofs use variational arguments to deduce rates in the fractional Sobolev norm from energy estimates between the fractional and the standard Dirichlet energy.

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Leon Bungert, Félix del Teso. 2024-08-06. Convergence rates of the fractional to the local Dirichlet problem. https://doi.org/10.1016/j.jde.2026.114173

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