Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913494266281984 |
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| author | Caravenna, Laura Marconi, Elio Pinamonti, Andrea |
| author_facet | Caravenna, Laura Marconi, Elio Pinamonti, Andrea |
| contents | We prove Hölder regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For example, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost every point $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to 0$. Due to Lipschitz regularity along characteristics, this implies that at almost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) = o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14518 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group Caravenna, Laura Marconi, Elio Pinamonti, Andrea Analysis of PDEs 53C17, 35L60 We prove Hölder regularity of any continuous solution $u$ to a $1$-D scalar balance law $u_t + [f(u)]_x = g$, when the source term $g$ is bounded and the flux $f$ is nonlinear of order $\ell \in \mathbb{N}$ with $\ell \ge 2$. For example, $\ell = 3$ if $f(u) = u^3$. Moreover, we prove that at almost every point $(t,x)$, it holds $u(t,x+h) - u(t,x) = o(|h|^{\frac{1}{\ell}})$ as $h \to 0$. Due to Lipschitz regularity along characteristics, this implies that at almost every point $(t,x)$, it holds $u(t+k,x+h) - u(t,x) = o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply the results to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group. |
| title | Hölder regularity of continuous solutions to balance laws and applications in the Heisenberg group |
| topic | Analysis of PDEs 53C17, 35L60 |
| url | https://arxiv.org/abs/2311.14518 |