Comparison Principles for the Finsler Infinity Laplacian with Applications to Minimal Lipschitz Extensions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911871477481472 |
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| author | Morfe, Peter S. |
| author_facet | Morfe, Peter S. |
| contents | This paper proves comparison principles for elliptic PDE involving the Finsler infinity Laplacian, a second-order differential operator with discontinuities in the gradient variable arising in $L^{\infty}$-variational problems and tug-of-war games. The core of the paper consists in proving generalized cone comparison principles. Among other consequences, these results imply that, for any Finsler norm $φ$ in $\mathbb{R}^{d}$, a function $u$ is a $φ$-absolutely minimizing Lipschitz extension if and only if it is a viscosity solution of the $φ$-infinity Laplace equation, settling a longstanding question in the $L^{\infty}$-calculus of variations. The proofs combine new geometric constructions with classical notions from convex analysis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Comparison Principles for the Finsler Infinity Laplacian with Applications to Minimal Lipschitz Extensions Morfe, Peter S. Analysis of PDEs This paper proves comparison principles for elliptic PDE involving the Finsler infinity Laplacian, a second-order differential operator with discontinuities in the gradient variable arising in $L^{\infty}$-variational problems and tug-of-war games. The core of the paper consists in proving generalized cone comparison principles. Among other consequences, these results imply that, for any Finsler norm $φ$ in $\mathbb{R}^{d}$, a function $u$ is a $φ$-absolutely minimizing Lipschitz extension if and only if it is a viscosity solution of the $φ$-infinity Laplace equation, settling a longstanding question in the $L^{\infty}$-calculus of variations. The proofs combine new geometric constructions with classical notions from convex analysis. |
| title | Comparison Principles for the Finsler Infinity Laplacian with Applications to Minimal Lipschitz Extensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.05684 |