Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914259296845824 |
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| author | Quaas, Alexander Topp, Erwin |
| author_facet | Quaas, Alexander Topp, Erwin |
| contents | In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $Ω\subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on $\partial Ω$. We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11241 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions Quaas, Alexander Topp, Erwin Analysis of PDEs In this article, we study the ergodic problem associated to viscous Hamilton-Jacobi equation where the diffusion is governed by the censored fractional Laplacian, a nonlocal elliptic operator restricted to a bounded domain $Ω\subset \mathbb{R}^N$. We restrict ourselves to the case in which the nonlinear gradient term has a scaling less or equal than the fractional order of the diffusion. In similarity to its second-order counterpart, we provide existence of ergodic pairs involving solutions that blow-up on $\partial Ω$. We use the celebrated vanishing discount method, where the analysis of the approximated solutions have its own interest, leading to qualitative properties for the ergodic problem such as precise blow-up rates for the solution and characterization of the ergodic constant. The main difficulties arise from the state-dependency of the operator, from which the arguments of the local case based on well-known invariance properties of the Laplacian are not longer at disposal. |
| title | Ergodic pairs for fractional Hamilton-Jacobi equations on bounded domains: large solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.11241 |