Gradient bounds for viscosity solutions to certain elliptic equations
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908626317213696 |
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| author | Jeffres, Thalia Li, Xiaolong |
| author_facet | Jeffres, Thalia Li, Xiaolong |
| contents | Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerate elliptic equation \( F(x, u, \nabla u, D^{2} u) = 0 \) in the viscosity sense. The equations under consideration here have second-order terms of the form \( -{\rm Trace} \, (\mathcal{A} (\|\nabla u \|) \cdot D^{2} u) , \) where \( \mathcal{A} \) is an \( n\times n\) matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02073 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gradient bounds for viscosity solutions to certain elliptic equations Jeffres, Thalia Li, Xiaolong Analysis of PDEs 35J60, 35D40, 35B10, 35B05 Our principal object of study is the modulus of continuity of a periodic or uniformly vanishing function \( u: \mathbb{R} ^{n} \rightarrow \mathbb{R} \) which satisfies a degenerate elliptic equation \( F(x, u, \nabla u, D^{2} u) = 0 \) in the viscosity sense. The equations under consideration here have second-order terms of the form \( -{\rm Trace} \, (\mathcal{A} (\|\nabla u \|) \cdot D^{2} u) , \) where \( \mathcal{A} \) is an \( n\times n\) matrix which is symmetric and positive semi-definite. Following earlier work, \cite{Li21}, of the second author, which addressed the parabolic case, we identify a one-dimensional equation for which the modulus of continuity is a subsolution. In favorable cases, this one-dimensional operator can be used to derive a gradient bound on $u$ or to draw other conclusions about the nature of the solution. |
| title | Gradient bounds for viscosity solutions to certain elliptic equations |
| topic | Analysis of PDEs 35J60, 35D40, 35B10, 35B05 |
| url | https://arxiv.org/abs/2511.02073 |