Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911452289302528 |
|---|---|
| author | Korte, Riikka Rogovin, Sari Shanmugalingam, Nageswari Takala, Timo |
| author_facet | Korte, Riikka Rogovin, Sari Shanmugalingam, Nageswari Takala, Timo |
| contents | Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincaré inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincaré inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15701 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques Korte, Riikka Rogovin, Sari Shanmugalingam, Nageswari Takala, Timo Analysis of PDEs Metric Geometry 31E05, 30L10, 30L15, 51F30 Within the setting of metric spaces equipped with a doubling measure and supporting a $p$-Poincaré inequality, establishing existence of solutions to Dirichlet problem in a bounded domain in such a metric space is accomplished via direct methods of calculus of variation and the use of a Maz'ya type inequality, which is a consequence of the Poincaré inequality. However, when the domain and its boundary are unbounded, such a method is unavailable. In this paper, using the technique of sphericalization developed in the prior paper~[32], we establish the existence of solutions to the Dirichlet boundary value problem for $p$-harmonic functions in unbounded uniform domains with unbounded boundary when $1<p<\infty$. We also explore the issue of whether such solutions are unique by considering $p$-parabolicity and $p$-hyperbolicity properties of the domain. |
| title | Solving Dirichlet problem on unbounded uniform domains by using sphericalization techniques |
| topic | Analysis of PDEs Metric Geometry 31E05, 30L10, 30L15, 51F30 |
| url | https://arxiv.org/abs/2602.15701 |