Mean oscillation conditions for nonlinear equation and regularity results
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908836453941248 |
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| author | Hästö, Peter Lee, Mikyoung Ok, Jihoon |
| author_facet | Hästö, Peter Lee, Mikyoung Ok, Jihoon |
| contents | We consider general nonlinear elliptic equations of the form \[ \operatorname{div}\, A(x,Du) = 0 \quad \text{in } Ω, \] where $A:Ω\times \mathbb R^n \to \mathbb R^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent to the point-wise uniform ellipticity of $A$. We establish sharp and comprehensive mean oscillation conditions on $A(x,ξ)$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,s}$-regularity results. The results provide new conditions even in the standard $p$-growth case with coefficient $\operatorname{div}(a(x)|Du|^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02159 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean oscillation conditions for nonlinear equation and regularity results Hästö, Peter Lee, Mikyoung Ok, Jihoon Analysis of PDEs 35J62, 35B65, 35R05, 35J92, 46E35 We consider general nonlinear elliptic equations of the form \[ \operatorname{div}\, A(x,Du) = 0 \quad \text{in } Ω, \] where $A:Ω\times \mathbb R^n \to \mathbb R^n$ satisfies a quasi-isotropic $(p,q)$-growth condition, which is equivalent to the point-wise uniform ellipticity of $A$. We establish sharp and comprehensive mean oscillation conditions on $A(x,ξ)$ with respect to the $x$ variable to obtain $C^1$- and $W^{1,s}$-regularity results. The results provide new conditions even in the standard $p$-growth case with coefficient $\operatorname{div}(a(x)|Du|^{p-2}Du)=0$. Also included are variable exponent growth with and without perturbation as well as borderline double-phase growth and double-phase growth with a coefficient. |
| title | Mean oscillation conditions for nonlinear equation and regularity results |
| topic | Analysis of PDEs 35J62, 35B65, 35R05, 35J92, 46E35 |
| url | https://arxiv.org/abs/2504.02159 |