Regularity theory of fully nonlinear anisotropic equations with nonstandard growth

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Regularity theory of fully nonlinear anisotropic equations with nonstandard growth

김한나 0 3144
구분 박사학위 논문 발표
일정 2026-06-12(금) 15:00~16:00
세미나실 27동 220호
강연자 김홍수 (서울대학교)
담당교수 변순식
기타
We establish a regularity theory for a general class of fully nonlinear anisotropic equations with nonstandard growth in a nondivergence setting.

First, we prove Lipschitz regularity for viscosity solutions of anisotropic equations with nonstandard growth, without imposing any restriction on the gap between the highest and lowest growth exponents.
The primary models for our equations are the anisotropic $(p_i)$-Laplacian with H\"older coefficients.
Our proof is based on an anisotropic variant of the seminal Ishii–Lions method.

Second, we prove Aleksandrov-Bakelman-Pucci estimates and Harnack inequalities for viscosity solutions of anisotropic-$p$-Laplacian equations with $L^n$ data.
Our main approach is an adaptation of the sliding paraboloid method with the suitable anisotropic test functions and a method of reduction to a slice where the test function becomes singular.

Finally, we prove Harnack inequalities for viscosity solutions of anisotropic equations with nonstandard growth with suitable closeness assumption on the exponents.
The primary models for our equations are anisotropic $(p_i)$-Laplacian with rough coefficients.
A central contribution is the explicit construction of a novel barrier function and adaption of the intrinsic geometry techniques. 

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