Harnack inequality for non-uniformly elliptic equations in non-divergence form
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913032830976000 |
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| author | Bowman, David |
| author_facet | Bowman, David |
| contents | We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge λ(x) I$ and the only assumption is that $λ^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13303 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Harnack inequality for non-uniformly elliptic equations in non-divergence form Bowman, David Analysis of PDEs 35J70, 35J15, 35D40, 35B65, 35B50 We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge λ(x) I$ and the only assumption is that $λ^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$. |
| title | Harnack inequality for non-uniformly elliptic equations in non-divergence form |
| topic | Analysis of PDEs 35J70, 35J15, 35D40, 35B65, 35B50 |
| url | https://arxiv.org/abs/2604.13303 |