Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations

2023/02/13 by Priyank Oza, Oza, Priyank, Jagmohan Tyagi +1 · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems

paper · pdf · doi:10.48550/arxiv.2302.06046

Abstract

We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, C:=\x: Du(x)=0\. Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, s∈((1)/(2),1). Due to inapplicability of comparison principle for the equations under consideration, one can not use the classical Perron's method for the existence of a solution. However, using the Lipschitz estimates established in theorem and vanishing viscosity method, we get the existence of solution. We further prove interior C^1,δ regularity of the viscosity solutions using an improvement of the flatness technique when s is close enough to 1.

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• Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term
• Mixed fully nonlinear local and nonlocal elliptic operators in Heisenberg group

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