Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913201428365312 |
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| author | Oza, Priyank Tyagi, Jagmohan |
| author_facet | Oza, Priyank Tyagi, Jagmohan |
| contents | We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, $s\in\big(\frac{1}{2},1\big).$ 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 $1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06046 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations Oza, Priyank Tyagi, Jagmohan Analysis of PDEs 35B65, 35J70, 35J60, 35D40, 47G20 We consider a class of variable-exponent mixed fully nonlinear local and nonlocal degenerate elliptic equations, which degenerate along the set of critical points, $C:=\big\{x:\,Du(x)=0\big\}.$ Under general conditions, first, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, $s\in\big(\frac{1}{2},1\big).$ 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 $1.$ |
| title | Regularity of solutions to variable-exponent degenerate mixed fully nonlinear local and nonlocal equations |
| topic | Analysis of PDEs 35B65, 35J70, 35J60, 35D40, 47G20 |
| url | https://arxiv.org/abs/2302.06046 |