Universal regularity estimates for solutions to fully nonlinear elliptic equations with oblique boundary data

2024/02/27 by Bessa, Junior da S., da Silva, João Vitor, Ricarte, Gleydson C.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.17899

Abstract

In this work, we establish universal moduli of continuity for viscosity solutions to fully nonlinear elliptic equations with oblique boundary conditions, whose general model is given by \ F(D²u,x) · amp;= · amp; f(x) in Ω
β(x) ⋅ Du(x) + γ(x) u(x) · amp;= · amp; g(x) on ∂ Ω. . Such regularity estimates are achieved by exploring the integrability properties of f based on different scenarios, making a VMO assumption on the coefficients of F, and by considering suitable smoothness properties on the boundary data β, γ and g. Particularly, we derive sharp estimates for borderline cases where f ∈ Lⁿ(Ω) and f∈ p-\textrmBMO(Ω). Additionally, for source terms in L^p(Ω), for p ∈ (n, ∞), we obtain sharp gradient estimates. Finally, we also address Schauder-type estimates for convex/concave operators and suitable Hölder data.

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