Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866913089140555776 |
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| author | Peres, Yuval Wang, Han |
| author_facet | Peres, Yuval Wang, Han |
| contents | We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ with a $\{0,1\}$-valued function as the boundary condition and study the dependence of the boundary derivative on $p$ as $p\downarrow1$. We provide sufficient conditions for the derivative to explode at rate $\frac{C_Ω}{p-1}$ and to decay at rate $\exp(-\frac{c_Ω}{p-1})$. Surprisingly, whether explosion or decay occurs is not determined locally. We also present a critical example of a cylinder where this derivative explodes at rate $\frac{C_d}{\sqrt{p-1}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality Peres, Yuval Wang, Han Analysis of PDEs Probability 35J92, 91A15 We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$ with a $\{0,1\}$-valued function as the boundary condition and study the dependence of the boundary derivative on $p$ as $p\downarrow1$. We provide sufficient conditions for the derivative to explode at rate $\frac{C_Ω}{p-1}$ and to decay at rate $\exp(-\frac{c_Ω}{p-1})$. Surprisingly, whether explosion or decay occurs is not determined locally. We also present a critical example of a cylinder where this derivative explodes at rate $\frac{C_d}{\sqrt{p-1}}$. |
| title | Explosion versus decay for boundary derivatives of $p$-harmonic functions as $p$ tends to 1: nonlocality |
| topic | Analysis of PDEs Probability 35J92, 91A15 |
| url | https://arxiv.org/abs/2605.03322 |