2025/06/03 by Josep M. Gallegos, Gallegos, Josep M. · 1 citation
Mathematics · #Nonlinear Partial Differential Equations #Geometric Analysis and Curvature Flows #Analytic and geometric function theory
paper · pdf · doi:10.48550/arxiv.2506.03431
Let Ω⊂ \mathbb Rn+1, n≥1, be a bounded open set satisfying the interior corkscrew condition with a uniformly n-rectifiable boundary but without any connectivity assumptions. We establish the estimate \Vert ∂νuf \VertM \lesssim \Vert ∇H f \VertL1(∂Ω), for all f\inLip(∂Ω) where uf is the solution to the Dirichlet problem with boundary data f, ∂νuf is the normal derivative of uf at the boundary in the weak sense, \Vert ⋅ \VertM denotes the total variation norm and ∇H f is the Hajłasz-Sobolev gradient of f. Conversely, if Ω⊂ \mathbb Rn+1 is a corkscrew domain with n-Ahlfors regular boundary and the previous inequality holds for solutions to the Dirichlet problem on Ω, then ∂Ω must satisfy the weak-no-boxes condition introduced by David and Semmes. Hence, in the planar case, the one-sided Rellich inequality characterizes the uniform rectifiability of ∂Ω. We also show solvability of the regularity problem in weak L1 for bounded corkscrew domains with a uniformly n-rectifiable boundary, that is \Vert N(∇ uf) \VertL1,∞(∂Ω) \lesssim \Vert ∇H f\VertL1(∂Ω), for all f\inLip(∂Ω) where N is the nontangential maximal operator. As an application of our results, we prove that for general elliptic operators, the solvability of the Dirichlet problem does not imply the solvability of the regularity problem.