We consider the homogeneous Dirichlet problem for the integral fractional Laplacian. We prove optimal Sobolev regularity estimates in Lipschitz domains satisfying an exterior ball condition. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

翻译：我们考虑的是集成分数拉普拉西亚的同质 Dirichlet 问题。 在利普西茨域,我们证明最理想的 Sobolev 常规性估计符合外球条件。 我们用贪婪的算法来提出分级的细片介质的构造,并得出近似最佳的近似趋同率,通过连续的片断线函数来解决这个问题。 非线性索博列夫比例表决定了规律性和近似性之间的关系。