Carleson perturbations of locally Lipschitz elliptic operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913973766455296 |
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| author | Feneuil, Joseph |
| author_facet | Feneuil, Joseph |
| contents | In one-sided Chord-Arc Domains $Ω$, we demonstrate that the $A_\infty$-absolute continuity of the elliptic measure with respect to the surface measure remains stable under $L^2$ Carleson perturbations. This stability holds provided that either the elliptic operator $L_0=-\operatorname{div} A_0\nabla$, which is being perturbed, or the perturbed operator $L_1=-\operatorname{div} A_1\nabla$ satisfies the condition $\sup_{X\in Ω}\operatorname{dist}(X,\partial Ω)|\nabla A_i(X)| <\infty$ on its coefficients. $L^2$ Carleson perturbations are slightly more general than those previously discussed in the literature.
The proof hinges on the availability of a comprehensive elliptic theory and a domain $Ω$ that allows uniform non-tangential access to any point on its boundary. Consequently, while the current theory of $L^2$ Carleson perturbations can be extended to more general contexts, we have chosen not to do so in order to simplify the presentation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Carleson perturbations of locally Lipschitz elliptic operators Feneuil, Joseph Analysis of PDEs 35J25 In one-sided Chord-Arc Domains $Ω$, we demonstrate that the $A_\infty$-absolute continuity of the elliptic measure with respect to the surface measure remains stable under $L^2$ Carleson perturbations. This stability holds provided that either the elliptic operator $L_0=-\operatorname{div} A_0\nabla$, which is being perturbed, or the perturbed operator $L_1=-\operatorname{div} A_1\nabla$ satisfies the condition $\sup_{X\in Ω}\operatorname{dist}(X,\partial Ω)|\nabla A_i(X)| <\infty$ on its coefficients. $L^2$ Carleson perturbations are slightly more general than those previously discussed in the literature. The proof hinges on the availability of a comprehensive elliptic theory and a domain $Ω$ that allows uniform non-tangential access to any point on its boundary. Consequently, while the current theory of $L^2$ Carleson perturbations can be extended to more general contexts, we have chosen not to do so in order to simplify the presentation. |
| title | Carleson perturbations of locally Lipschitz elliptic operators |
| topic | Analysis of PDEs 35J25 |
| url | https://arxiv.org/abs/2408.10061 |