Stable solutions to semilinear elliptic equations for operators with variable coefficients

2022/06/03 by Iñigo U. Erneta, Erneta, Iñigo U. · 1 citation
Computer Science · Mathematics · #35B35 #35B45 #35B65 #35J15 #35J61 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.01572

openalex publication_date 2022/06/03 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28

Abstract

In this paper we extend the interior regularity results for stable solutions in [Cabré, Figalli, Ros-Oton, and Serra, Acta Math. 224 (2020)] to operators with variable coefficients. We show that stable solutions to the semilinear elliptic equation a_ij(x)u_ij + b_i(x) u_i + f(u) = 0 are Hölder continuous in the optimal range of dimensions n ≤ 9. Our bounds are independent of the nonlinearity f ∈ C¹, which we assume to be non-negative. The main achievement of our work is to make the constants in our estimates depend on the C¹ norm of a_ij and the C⁰ norm of b_i, instead of their C² and C¹ norms, respectively, which arise in a first approach to the computations.

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