The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

By S. Hofmann, Carlos E. Kenig, Svitlana Mayboroda, Jill Pipher. 2013

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The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $$t$$t-independent complex bounded measurable coefficients ($$t$$t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $$L^{p'}$$Lp′, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $$L^p$$Lp. Moreover, the solutions admit layer potential representations. In particular, we prove that for any elliptic operator with $$t$$t-independent real (possibly non-symmetric) coefficients there exists a $$p>1$$p>1 such that the Regularity problem is well-posed in $$L^p$$Lp.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a18d88dba5518fb764c57e70f0b475b9e644aaa8