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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.06487 |
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| _version_ | 1866913396088111104 |
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| author | Torres-Latorre, Clara |
| author_facet | Torres-Latorre, Clara |
| contents | We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing for the first time a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side $f \in L^q$ for $q > n+2$. In the case of the heat equation, we also show the optimal $C^{1-\varepsilon}$ regularity of the quotient.
As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are $C^{1,α}$ in the parabolic obstacle problem and in the parabolic Signorini problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06487 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parabolic boundary Harnack inequalities with right-hand side Torres-Latorre, Clara Analysis of PDEs 35K10 (Primary), 35R35 (Secondary) We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing for the first time a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side $f \in L^q$ for $q > n+2$. In the case of the heat equation, we also show the optimal $C^{1-\varepsilon}$ regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are $C^{1,α}$ in the parabolic obstacle problem and in the parabolic Signorini problem. |
| title | Parabolic boundary Harnack inequalities with right-hand side |
| topic | Analysis of PDEs 35K10 (Primary), 35R35 (Secondary) |
| url | https://arxiv.org/abs/2307.06487 |