Faustmann, M., Marcati, C., Melenk, J. M., & Schwab, C. (2025). Weighted analytic regularity for the integral fractional Laplacian in polyhedra. Analysis and Applications, 23(4), 511–552. https://doi.org/10.1142/S0219530524500386
On polytopal domains in ℝ3, we prove weighted analytic regularity of solutions to the Dirichlet problem for the shifted integral fractional Laplacian with analytic right-hand side. Employing the Caffarelli-Silvestre extension allows one to localize the problem and to decompose the regularity estimates into results on vertex, edge, face, vertex-edge, vertex-face, edge-face and vertex-edge-face neighborhoods of the boundary. Using tangential differentiability of the extended solutions, a bootstrapping argument based on Caccioppoli inequalities on dyadic decompositions of the neighborhoods provides weighted, analytic control of higher-order solution derivatives.
