Fully Nonlinear Evolution equations: From Critical Integrability to Schauder Estimates

2026/08/02 by Junior da Silva Bessa, José Erivamberto L. Oliveira, Patrícia Renata Pereira Regis
Mathematics · #math.AP

paper · pdf

30 pages

arxiv created 2026/08/02 · arxiv updated 2026/08/04

Abstract

We investigate the sharp regularity up to the boundary for viscosity solutions of fully nonlinear parabolic equations with oblique derivative boundary conditions given by \ F(D²u,x,t) - u_t · = · f(x,t) · in · Q₁⁺,
β(x,t) ⋅ Du · = · g(x,t) · on · Q₁^*. . The regularity theory is developed according to the integrability of the source term, the smoothness of the boundary data, and the oscillation of the coefficients of the operator \(F\), using a compactness method combined with polynomial approximation. In the borderline case \(f∈ Lⁿ⁺²\), we obtain Log-Lipschitz continuity of solutions up to the boundary. Under stronger integrability, specifically when \(f∈ L^p\) for some \(p>n+2\), we establish optimal \(C^{1+α', (1+α')/(2)}\) boundary estimates. At a higher regularity level, we prove Schauder-type estimates under appropriate assumptions on the operator \(F\) and the boundary data. As a byproduct, we obtain parabolic \(C^1,Log-Lip\) regularity in the critical borderline case where the source term belongs to BMO space.

Citations

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