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- W4287754227 abstract "In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-varepsilon < p<frac{2(n-1)}{(n-3)}+varepsilon$, for systems in dimension $n=2,3$ in the range $2-varepsilon < p<infty$. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in $L^p$) in the range $2-varepsilon < p<2+varepsilon$. Secondly, we look at perturbation type-results where we can deduce solvability of the $L^p$ Dirichlet problem for one operator from known $L^p$ Dirichlet solvability of a lqlq close operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper arXiv:1708.02289; we establish solvability of the $L^p$ Dirichlet problem in the interval $2-varepsilon < p<frac{2(n-1)}{(n-2)}+varepsilon$ under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lam'e operator for isotropic inhomogeneous materials with Poisson ratio $nu<0.396$. In this specific case further improvements of the solvability range are possible, see the upcoming work with J. Li and J. Pipher." @default.
- W4287754227 created "2022-07-26" @default.
- W4287754227 creator A5074858336 @default.
- W4287754227 date "2020-06-21" @default.
- W4287754227 modified "2023-10-18" @default.
- W4287754227 title "The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lam'e system" @default.
- W4287754227 doi "https://doi.org/10.48550/arxiv.2006.13015" @default.
- W4287754227 hasPublicationYear "2020" @default.
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