Pointwise regularity for locally uniformly elliptic equations and applications
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866916243201589248 |
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| author | Lian, Yuanyuan Zhang, Kai |
| author_facet | Lian, Yuanyuan Zhang, Kai |
| contents | In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,α}$ ($k\geq 1$, $0<α<1$) regularity with smallness assumptions on the solution and the right-hand term. As applications, we obtain various interior pointwise regularity for several classical elliptic equations, i.e., the prescribed mean curvature equation, the Monge-Ampère equation, the $k$-Hessian equations, the $k$-Hessian quotient equations and the Lagrangian mean curvature equation. Moreover, the smallness assumptions are necessary in most cases (Remark 2.6, Remark 3.5, Remark 4.7, Remark 5.4 and Remark 6.5). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_07199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pointwise regularity for locally uniformly elliptic equations and applications Lian, Yuanyuan Zhang, Kai Analysis of PDEs 35B65, 35D40, 35J60, 35J96, 35J93 In this paper, we study the regularity for viscosity solutions of locally uniformly elliptic equations and obtain a series of interior pointwise $C^{k,α}$ ($k\geq 1$, $0<α<1$) regularity with smallness assumptions on the solution and the right-hand term. As applications, we obtain various interior pointwise regularity for several classical elliptic equations, i.e., the prescribed mean curvature equation, the Monge-Ampère equation, the $k$-Hessian equations, the $k$-Hessian quotient equations and the Lagrangian mean curvature equation. Moreover, the smallness assumptions are necessary in most cases (Remark 2.6, Remark 3.5, Remark 4.7, Remark 5.4 and Remark 6.5). |
| title | Pointwise regularity for locally uniformly elliptic equations and applications |
| topic | Analysis of PDEs 35B65, 35D40, 35J60, 35J96, 35J93 |
| url | https://arxiv.org/abs/2405.07199 |