Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909649192615936 |
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| author | Lundström, Niklas L. P. Olofsson, Marcus Singh, Jesper |
| author_facet | Lundström, Niklas L. P. Olofsson, Marcus Singh, Jesper |
| contents | We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to vanish near the boundary and permits general, possibly unbounded, lower-order terms. A key consequence is the derivation of boundary Harnack inequalities for a broad class of fully nonlinear, nonhomogeneous equations near $C^{1,1}$-boundaries.
In combination with $C^{1,α}$-estimates, we also obtain that quotients of positive vanishing solutions are Hölder continuous near $C^{1,1}$-boundaries.This result applies to a wide family of fully nonlinear uniformly elliptic PDEs; and for $p(x)$-harmonic functions and planar $\infty$-harmonic functions near locally flat boundaries. We end by deriving some Phragmén-Lindelöf-type corollaries in unbounded domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_12477 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries Lundström, Niklas L. P. Olofsson, Marcus Singh, Jesper Analysis of PDEs 35J60, 35J70, 35B40, 35B53, 35D40, 35J25, 35J92, 42B37 We reduce the problem of proving decay estimates for viscosity solutions of fully nonlinear PDEs to proving analogous estimates for solutions of one-dimensional ordinary differential inequalities. Our machinery allow the ellipticity to vanish near the boundary and permits general, possibly unbounded, lower-order terms. A key consequence is the derivation of boundary Harnack inequalities for a broad class of fully nonlinear, nonhomogeneous equations near $C^{1,1}$-boundaries. In combination with $C^{1,α}$-estimates, we also obtain that quotients of positive vanishing solutions are Hölder continuous near $C^{1,1}$-boundaries.This result applies to a wide family of fully nonlinear uniformly elliptic PDEs; and for $p(x)$-harmonic functions and planar $\infty$-harmonic functions near locally flat boundaries. We end by deriving some Phragmén-Lindelöf-type corollaries in unbounded domains. |
| title | Estimates for viscosity solutions of fully nonlinear equations near smooth boundaries |
| topic | Analysis of PDEs 35J60, 35J70, 35B40, 35B53, 35D40, 35J25, 35J92, 42B37 |
| url | https://arxiv.org/abs/2506.12477 |