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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.07993 |
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| _version_ | 1866909287848083456 |
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| author | Liang, Jingqi |
| author_facet | Liang, Jingqi |
| contents | In this paper, we obtain $C^{1}$ and $C^{1,1}$ regularity of $L^{n}$-viscosity solutions for general semilinear elliptic equation in nondivergence form under some more weaker assumptions, which generalize the result for equations with nonhomogeneous term $f(x)$ to $f(x,u)$. In particular, the nonhomogeneous term $f(x,u)$ is assumed optimally to satisfy unform Dini continuity condition in $u$ and modified $C^{1,1}$ Newtonian potential condition in $x$. For unbounded coefficients, if $a_{ij}$ is $C_{n}^{-1,1}$ at $x_{0}\inΩ$ with small modulus, $b_{i}\in L^{q}(Ω)$ for some $q>n$, the solution is $C^{1}$ at $x_{0}$. Furthermore, if $a_{ij},~b_{i}$ are Dini continuous at $x_{0}$, the solution is $C^{1,1}$ at $x_{0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_07993 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interior pointwise $C^{1}$ and $C^{1,1}$ regularity of solutions for general semilinear elliptic equation in nondivergence form Liang, Jingqi Analysis of PDEs In this paper, we obtain $C^{1}$ and $C^{1,1}$ regularity of $L^{n}$-viscosity solutions for general semilinear elliptic equation in nondivergence form under some more weaker assumptions, which generalize the result for equations with nonhomogeneous term $f(x)$ to $f(x,u)$. In particular, the nonhomogeneous term $f(x,u)$ is assumed optimally to satisfy unform Dini continuity condition in $u$ and modified $C^{1,1}$ Newtonian potential condition in $x$. For unbounded coefficients, if $a_{ij}$ is $C_{n}^{-1,1}$ at $x_{0}\inΩ$ with small modulus, $b_{i}\in L^{q}(Ω)$ for some $q>n$, the solution is $C^{1}$ at $x_{0}$. Furthermore, if $a_{ij},~b_{i}$ are Dini continuous at $x_{0}$, the solution is $C^{1,1}$ at $x_{0}$. |
| title | Interior pointwise $C^{1}$ and $C^{1,1}$ regularity of solutions for general semilinear elliptic equation in nondivergence form |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2408.07993 |