Unique continuation on planar graphs
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911145979281408 |
|---|---|
| author | Bou-Rabee, Ahmed Cooperman, William Ganguly, Shirshendu |
| author_facet | Bou-Rabee, Ahmed Cooperman, William Ganguly, Shirshendu |
| contents | We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13728 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unique continuation on planar graphs Bou-Rabee, Ahmed Cooperman, William Ganguly, Shirshendu Analysis of PDEs Probability We show that a discrete harmonic function which is bounded on a large portion of a periodic planar graph is constant. A key ingredient is a new unique continuation result for the weighted graph Laplacian. The proof relies on the structure of level sets of discrete harmonic functions, using arguments as in Bou-Rabee--Cooperman--Dario (2023) which exploit the fact that, on a planar graph, the sub- and super-level sets cannot cross over each other. In the special case of the square lattice this yields a new, geometric proof of the Liouville theorem of Buhovsky--Logunov--Malinnikova--Sodin (2017). |
| title | Unique continuation on planar graphs |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2309.13728 |