Global Sobolev regularity for nonvariational operators built with homogeneous Hörmander vector fields
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866909178599047168 |
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| author | Biagi, Stefano Bramanti, Marco |
| author_facet | Biagi, Stefano Bramanti, Marco |
| contents | We consider a class of nonvariational degenerate elliptic operators of the kind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{ a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole $\mathbb{R}^{n}$ ($n>m$), possibly discontinuos but satisfying a $VMO$ assumption, and $X_{1},...,X_{m}$ are real smooth vector fields satisfying Hörmander rank condition in the whole $\mathbb{R}^{n}$ and $1$-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global $W_{X}^{2,p}$ a-priori estimates, for every $p\in\left( 1,\infty\right) $, of the kind: \[ \Vert u\Vert_{W_{X}^{2,p}(\mathbb{R}^{n})}\leq c\left\{ \left\Vert Lu\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }+\left\Vert u\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }\right\} \] for every $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) .$ We also prove higher order estimates and corresponding regularity results: if $a_{ij}\in W_{X}^{k,\infty}\left( \mathbb{R}^{n}\right) $, $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) $, $Lu\in W_{X}^{k,p}\left( \mathbb{R}^{n}\right) $, then $u\in W_{X}^{k+2,p}\left( \mathbb{R}^{n}\right) $ and \[ \Vert u\Vert_{W_{X}^{k+2,p}(\mathbb{R}^{n})}\leq c\left\{ \Vert Lu\Vert_{W_{X}^{k,p}(\mathbb{R}^{n})}+\Vert u\Vert_{L^{p}(\mathbb{R}^{n} )}\right\} . \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15367 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Global Sobolev regularity for nonvariational operators built with homogeneous Hörmander vector fields Biagi, Stefano Bramanti, Marco Analysis of PDEs 35B45, 35B65 (primary), 35J70, 35H10, 42B25 (secondary) We consider a class of nonvariational degenerate elliptic operators of the kind \[ Lu=\sum_{i,j=1}^{m}a_{ij}\left( x\right) X_{i}X_{j}u \] where $\left\{ a_{ij}\left( x\right) \right\} _{i,j=1}^{m}$ is a symmetric uniformly positive matrix of bounded measurable functions defined in the whole $\mathbb{R}^{n}$ ($n>m$), possibly discontinuos but satisfying a $VMO$ assumption, and $X_{1},...,X_{m}$ are real smooth vector fields satisfying Hörmander rank condition in the whole $\mathbb{R}^{n}$ and $1$-homogeneous w.r.t. a family of nonisotropic dilations. We do not assume that the vector fields are left invariant w.r.t. an underlying Lie group of translations. We prove global $W_{X}^{2,p}$ a-priori estimates, for every $p\in\left( 1,\infty\right) $, of the kind: \[ \Vert u\Vert_{W_{X}^{2,p}(\mathbb{R}^{n})}\leq c\left\{ \left\Vert Lu\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }+\left\Vert u\right\Vert _{L^{p}\left( \mathbb{R}^{n}\right) }\right\} \] for every $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) .$ We also prove higher order estimates and corresponding regularity results: if $a_{ij}\in W_{X}^{k,\infty}\left( \mathbb{R}^{n}\right) $, $u\in W_{X}^{2,p}\left( \mathbb{R}^{n}\right) $, $Lu\in W_{X}^{k,p}\left( \mathbb{R}^{n}\right) $, then $u\in W_{X}^{k+2,p}\left( \mathbb{R}^{n}\right) $ and \[ \Vert u\Vert_{W_{X}^{k+2,p}(\mathbb{R}^{n})}\leq c\left\{ \Vert Lu\Vert_{W_{X}^{k,p}(\mathbb{R}^{n})}+\Vert u\Vert_{L^{p}(\mathbb{R}^{n} )}\right\} . \] |
| title | Global Sobolev regularity for nonvariational operators built with homogeneous Hörmander vector fields |
| topic | Analysis of PDEs 35B45, 35B65 (primary), 35J70, 35H10, 42B25 (secondary) |
| url | https://arxiv.org/abs/2312.15367 |