Boundary regularity of mixed local-nonlocal operators and its application

2022/04/15 by Biswas, Anup, Modasiya, Mitesh, Sen, Abhrojyoti · 7 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2204.07389

Abstract

Let Ω be a bounded C² domain in ℝⁿ and u∈ C(ℝⁿ) solves \beginaligned Δu + a Iu + C₀|Du| ≥ -K in Ω, Δu + a Iu - C₀|Du|≤ K in Ω, u=0 in Ω^c, \endaligned in the viscosity sense, where 0≤ a≤ A₀, C₀, K≥ 0, and I is a suitable nonlocal operator. We show that u/δ is in C^κ( Ω) for some κ∈ (0,1), where δ(x)=\rm dist(x, Ω^c). Using this result, we also establish that u∈ C^{1, γ}(Ω). Finally, we apply these results to study an overdetermined problem for mixed local-nonlocal operators.

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• Non-degeneracy and uniqueness of ground states to nonlinear elliptic equations with mixed local and nonlocal operators
• Optimal boundary regularity for mixed local and nonlocal equations
• Regularity for Mixed Local and Nonlocal Degenerate Elliptic Equations in the Heisenberg Group
• Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation
• Large solutions for subordinate spectral Laplacian

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