2025/04/14 by Massimo Lanza de Cristoforis, de Cristoforis, M. Lanza
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2504.11487
We consider a bounded open subset Ω of ℝn of class C1,α for some α∈]0,1[, and we define a distributional outward unit normal derivative for α-Hölder continuous solutions of the Helmholtz equation in the exterior of Ω that may not have a classical outward unit normal derivative at the boundary points of Ω and that may have an infinite Dirichlet integral around the boundary of Ω. Namely for solutions that do not belong to the classical variational setting. Then we show a Schauder boundary regularity result for α-Hölder continuous functions that have the Laplace operator in a Schauder space of negative exponent and we prove a uniqueness theorem for α-Hölder continuous solutions of the exterior Dirichlet and impedance boundary value problems for the Helmholtz equation that satisfy the Sommerfeld radiation condition at infinity in the above mentioned nonvariational setting.