Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910836264534016 |
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| author | Hou, Mingyi |
| author_facet | Hou, Mingyi |
| contents | We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An $L^1$-$L^p$ type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains Hou, Mingyi Analysis of PDEs 35K70, 35Q84, 35K65, 35B45, 35B09 (Primary) 35B65 (Secondary) We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An $L^1$-$L^p$ type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients. |
| title | Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains |
| topic | Analysis of PDEs 35K70, 35Q84, 35K65, 35B45, 35B09 (Primary) 35B65 (Secondary) |
| url | https://arxiv.org/abs/2407.00800 |