What type of study is this?

September 10, 2025

On global fractional Calderón–Zygmund regularity for the fractional Dirichlet problem on Lipschitz domains

Puntos clave

Weak solutions to the fractional Dirichlet problem exhibit global Sobolev regularity under specific conditions.
The findings include sharp global regularity estimates with a constant depending on domain and fractional parameters.
BMO-Sobolev regularity estimates are established for Lipschitz domains, linking weak solutions to their source functions.
Results imply that the BMO norm can't be improved to the L^ norm in bounded Lipschitz domains.

Resumen

Abstract Let n2, s (0, 1), and ⁿ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem equation* cases (-) ˢu=f \ \ & in\ \, \\ u=0 \ \ & in\ \ Rⁿ. cases equation* Precisely, we prove that there exists a positive constant (0, s] depending on n, s, and the Lipschitz constant of Ω such that, for any t, \1+, 2s\), when f Lq () with some q (n2s-t, , the weak solution u satisfies equation*\|u\|ₖ^ₓ, ₏ (Rⁿ) C\|f\|₋ₐ () equation* for all p1, 1t-). In particular, when Ω is a bounded C 1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with =s and they are sharp in this case. Moreover, if Ω is a bounded C^1, domain with (0, s) or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate equation*\| (-) ^s{2}u\|₁₌₎ (ₑ䂞) +\|^su\|₁₌₎ (ₑ䂞) C\|f\|₋ₐ () equation* for some q (ns, , which is sharp in the sense that the BMO norm can not be improved to the L^ norm.

Me gusta

Guardar