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Galerkin Approximation of the Fractional Sobolev Constant
Abstract
We establish sharp estimates for the discrete optimal constant of the fractional Sobolev inequality in dimension N≥ 1, with fractional exponent s∈ (0,min{1,N/2}). The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the problem is posed in the unit ball, for which we employ a quasi-uniform and regular mesh. Since the infimum defining the fractional Sobolev constant is not attained on bounded domains, these estimates quantify how far a piecewise affine mesh of size h can resolve the concentration of the optimizers: the discrete minimizers are close to bubbles concentrated at the scale h2(2−s)/(N+4(1−s)), and the energy gap is exactly of order h2(2−s)(N−2s)/(N+4(1−s)).
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BibTeX
@misc{dima2026galerkin,
title = {{Galerkin Approximation of the Fractional Sobolev Constant}},
author = {Andreea Dima and Liviu I. Ignat},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.13347},
eprint = {2605.13347},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.13347},
}
RIS
TY - UNPB
TI - Galerkin Approximation of the Fractional Sobolev Constant
AU - Andreea Dima
AU - Liviu I. Ignat
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2605.13347
UR - https://arxiv.org/abs/2605.13347
AB - We establish sharp estimates for the discrete optimal constant of the fractional Sobolev inequality in dimension N≥ 1, with fractional exponent s∈ (0,min{1,N/2}). The convergence rates that we establish take place for the Galerkin approximation with piecewise linear elements, when the problem is posed in the unit ball, for which we employ a quasi-uniform and regular mesh. Since the infimum defining the fractional Sobolev constant is not attained on bounded domains, these estimates quantify how far a piecewise affine mesh of size h can resolve the concentration of the optimizers: the discrete minimizers are close to bubbles concentrated at the scale h^2(2−s)/(N+4(1−s)), and the energy gap is exactly of order h^2(2−s)(N−2s)/(N+4(1−s)).
ER -