The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system

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Abstract

In the paper [11] we have introduced new solvability methodsforstrongly elliptic second order systems in divergence form on a domains abovea Lipschitz graph, satisfyingLp-boundary data forpnear 2. The main novelaspect of our result is that it applies to operators with coefficients of limitedregularity and applies to operators satisfying a natural Carleson condition thathas been first considered in the scalar case.In this paper we extend this result in several directions. Weimprove therange of solvability of theLpDirichlet problem to the interval 2−ε < p <2(n−1)(n−3)+ε, for systems in dimensionn= 2,3 in the range 2−ε < p <∞. Wedo this by considering solvability of the Regularity problem (with boundarydata having one derivative inLp) in the range 2−ε < p <2 +ε.Secondly, we look at perturbation type-results where we candeduce solv-ability of theLpDirichlet problem for one operator from knownLpDirichletsolvability of a “close” operator (in the sense of Carleson measure). This leadsto improvement of the main result of the paper [11]; we establish solvability oftheLpDirichlet problem in the interval 2−ε < p <2(n−1)(n−2)+εunder a muchweaker (oscillation-type) Carleson condition.A particular example of the system where all these results apply is the Lam ́eoperator for isotropic inhomogeneous materials with Poisson ratioν <0.396.In this specific case further improvements of the solvability range are possible,see [12].
Original languageEnglish
Number of pages41
JournalCommunications in partial differential equations
Publication statusAccepted/In press - 6 Feb 2021

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