The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients

Abstract

We establish solvability methods for strongly elliptic second order systems in divergence form with lower order (drift) terms on a domain above a Lipschitz graph, satisfying L p -boundary data for p near 2. The main novel aspect of our result is that the coefficients of the operator do not have to be constant or have very high regularity, instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A particular example of a system where this result can be applied is the Lamé operator for isotropic inhomogeneous materials.The systems case poses substantial new challenges not present in the scalar case. In particular, there is no maximum principle for general elliptic systems and De Giorgi -Nash -Moser theory is also not available. Despite this we are able to establish estimates for the square and nontangential maximal functions for the solution of the elliptic system and use these estimates to establish L p solvability for p near 2.

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

“…This paper is motivated by the known results concerning boundary value problems for second order elliptic equations in divergence form, when the coefficients satisfying a certain natural, minimal smoothness condition (refer [13], [18], [27]). It extends the results of the paper [11] in several interesting and important directions. Because of this we maintain as closely as possible the notation introduced in [11].…”

“…It extends the results of the paper [11] in several interesting and important directions. Because of this we maintain as closely as possible the notation introduced in [11].…”

“…We do this by considering solvability of the Regularity problem (with boundary data having one derivative in L p ) in the range 2 − ε < p < 2 + ε.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 "close" operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper [11]; we establish solvability of the L p Dirichlet problem in the interval 2 − ε < p < 2(n−1) (n−2) + ε under a much weaker (oscillation-type) Carleson condition.A particular example of the system where all these results apply is the Lamé operator for isotropic inhomogeneous materials with Poisson ratio ν < 0.396. In this specific case further improvements of the solvability range are possible, see [12].…”

“…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 "close" operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper [11]; we establish solvability of the L p Dirichlet problem in the interval 2 − ε < p < 2(n−1) (n−2) + ε under a much weaker (oscillation-type) Carleson condition.…”

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In the paper [11] 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.

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