The exterior Dirichlet problem for the homogeneous $k$-Hessian equation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910415333621760 |
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| author | Ma, Xi-Nan Zhang, Dekai |
| author_facet | Ma, Xi-Nan Zhang, Dekai |
| contents | We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_13504 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The exterior Dirichlet problem for the homogeneous $k$-Hessian equation Ma, Xi-Nan Zhang, Dekai Analysis of PDEs We study the exterior Dirichlet problem for the homogeneous $k$-Hessian equation. The prescribed asymptotic behavior at infinity of the solution is zero if $k<\frac{n}{2}$, it is $\log|x|+O(1)$ if $k=\frac{n}{2}$ and it is $|x|^{\frac{2k-n}{n}}+O(1)$ if $k>\frac{n}{2}$. By constructing smooth solutions of approximating non-degenerate $k$-Hessian equations with uniform $C^{1,1}$-estimates, we prove the existence part. The uniqueness follows from the comparison theorem and thus the $C^{1,1}$ regularity of the solution of the homogeneous $k$-Hessian equation in the exterior domain is proved. We also prove a uniform positive lower bound of the gradient. As an implication of the $C^{1,1}$ estimates, we derive an almost monotonicity formula along the level set of the approximating solution. In particular, we get an weighted geometric inequality which is a natural generalization of the $k=1$ case. |
| title | The exterior Dirichlet problem for the homogeneous $k$-Hessian equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2207.13504 |