Semi-local behaviour of non-local hypoelliptic equations: divergence form
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911874095775744 |
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| author | Loher, Amélie |
| author_facet | Loher, Amélie |
| contents | We derive the Strong Harnack inequality for a class of hypoelliptic integro-differential equations in divergence form. The proof is based on a priori estimates, and as such extends the first non-stochastic approach of the non-local parabolic Strong Harnack inequality by Kassmann-Weidner [arXiv:2303.05975] to hypoelliptic equations; however, in contrast to the parabolic case, we only obtain a semi-local result, in the sense that we require the equation to hold globally in velocity, which in particular does not contradict the counterexample constructed in [arXiv:2405.05223]. In a first step, we derive a local bound on the non-local tail on upper level sets by exploiting the coercivity of the cross terms. In a second step, we perform a De Giorgi argument in $L^1$, since we control the tail term only in $L^1$. This yields a linear $L^1$ to $L^\infty$ bound. Consequentially, we prove polynomial upper and exponential lower bounds on the fundamental solution by adapting Aronson's method to non-local hypoelliptic equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_05612 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semi-local behaviour of non-local hypoelliptic equations: divergence form Loher, Amélie Analysis of PDEs 45K05, 35H10, 35K99, 35B45, 35B65 We derive the Strong Harnack inequality for a class of hypoelliptic integro-differential equations in divergence form. The proof is based on a priori estimates, and as such extends the first non-stochastic approach of the non-local parabolic Strong Harnack inequality by Kassmann-Weidner [arXiv:2303.05975] to hypoelliptic equations; however, in contrast to the parabolic case, we only obtain a semi-local result, in the sense that we require the equation to hold globally in velocity, which in particular does not contradict the counterexample constructed in [arXiv:2405.05223]. In a first step, we derive a local bound on the non-local tail on upper level sets by exploiting the coercivity of the cross terms. In a second step, we perform a De Giorgi argument in $L^1$, since we control the tail term only in $L^1$. This yields a linear $L^1$ to $L^\infty$ bound. Consequentially, we prove polynomial upper and exponential lower bounds on the fundamental solution by adapting Aronson's method to non-local hypoelliptic equations. |
| title | Semi-local behaviour of non-local hypoelliptic equations: divergence form |
| topic | Analysis of PDEs 45K05, 35H10, 35K99, 35B45, 35B65 |
| url | https://arxiv.org/abs/2404.05612 |