Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918203303657472 |
|---|---|
| author | Björn, Anders Björn, Jana Manolis, David |
| author_facet | Björn, Anders Björn, Jana Manolis, David |
| contents | We study boundary regularity at the infinity point $\boldsymbol{\infty}$ for nonlinear elliptic equations of $p$-Laplace type in unbounded open sets $Ω\subset \mathbf{R}^n$. We consider the case $p \ge n \ge 2$ and characterize the regularity at $\boldsymbol{\infty}$ by means of Wiener-type integrals. Our approach uses circular inversion, which maps $\boldsymbol{\infty}$ to the origin and the original nonlinear equation to a similar weighted equation. The Wiener criterion at the origin for such equations is then transformed back to provide Wiener-type criteria at $\boldsymbol{\infty}$. When $p>n$, the criteria simplify so that $\boldsymbol{\infty}$ is regular if and only if the boundary $\partialΩ$ is unbounded. For $p=n$ this is not true, as shown by an example. This simplified criterion is also proved for $p$-harmonic functions in unbounded open subsets of Ahlfors $Q$-regular metric measure spaces with $Q<p$, supporting a Poincaré inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_12166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type Björn, Anders Björn, Jana Manolis, David Analysis of PDEs Primary: 35J25, 31C45, Secondary: 31E05 We study boundary regularity at the infinity point $\boldsymbol{\infty}$ for nonlinear elliptic equations of $p$-Laplace type in unbounded open sets $Ω\subset \mathbf{R}^n$. We consider the case $p \ge n \ge 2$ and characterize the regularity at $\boldsymbol{\infty}$ by means of Wiener-type integrals. Our approach uses circular inversion, which maps $\boldsymbol{\infty}$ to the origin and the original nonlinear equation to a similar weighted equation. The Wiener criterion at the origin for such equations is then transformed back to provide Wiener-type criteria at $\boldsymbol{\infty}$. When $p>n$, the criteria simplify so that $\boldsymbol{\infty}$ is regular if and only if the boundary $\partialΩ$ is unbounded. For $p=n$ this is not true, as shown by an example. This simplified criterion is also proved for $p$-harmonic functions in unbounded open subsets of Ahlfors $Q$-regular metric measure spaces with $Q<p$, supporting a Poincaré inequality. |
| title | Boundary regularity and Wiener-type criteria at infinity for nonlinear elliptic equations of $p$-Laplace type |
| topic | Analysis of PDEs Primary: 35J25, 31C45, Secondary: 31E05 |
| url | https://arxiv.org/abs/2511.12166 |