Interior pointwise C¹ and C^1,1 regularity of solutions for general semilinear elliptic equation in nondivergence form

2024/08/15 by Jingqi Liang, Liang, Jingqi
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2408.07993

openalex publication_date 2024/08/15 · openalex created_date 2025/01/03 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain C¹ and C^1,1 regularity of Lⁿ-viscosity solutions for general semilinear elliptic equation in nondivergence form under some more weaker assumptions, which generalize the result for equations with nonhomogeneous term f(x) to f(x,u). In particular, the nonhomogeneous term f(x,u) is assumed optimally to satisfy unform Dini continuity condition in u and modified C^1,1 Newtonian potential condition in x. For unbounded coefficients, if a_ij is C_n^-1,1 at x₀∈Ω with small modulus, b_i∈ L^q(Ω) for some q>n, the solution is C¹ at x₀. Furthermore, if a_ij,~b_i are Dini continuous at x₀, the solution is C^1,1 at x₀.

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