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Analyticity of layer potentials and L-2 solvability of boundary value problems for divergence form elliptic equations with complex L-infinity coefficients
N Dakota State University.
University Paris 11.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
University of Missouri.
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2011 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 226, no 5, 4533-4606 p.Article in journal (Refereed) Published
Abstract [en]

We consider divergence form elliptic operators of the form L = -div A (x)del, defined in Rn+1 = {(x, t) is an element of R-n x R}, n andgt;= 2, where the L-infinity coefficient matrix A is (n + 1) x (n + 1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L-2 (R-n) = L-2(partial derivative R-+(n+1)) is stable under complex, L-infinity perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L-2(R-n) whenever A (x) is real and symmetric (and thus, by our stability result, also when A is complex, parallel to A - A(0)parallel to(infinity) is small enough and A(0) is real, symmetric, L-infinity and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L-2 (resp. (L) over dot(1)(2)) data, for small complex perturbations of a real symmetric matrix. Previously, L-2 solvability results for complex (or even real but non-symmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A (j,n+1)= 0 = A(n+1,j), 1 andlt;= j andlt;= n, which corresponds to the Kato square root problem.

Place, publisher, year, edition, pages
Elsevier Science B.V. Amsterdam , 2011. Vol. 226, no 5, 4533-4606 p.
Keyword [en]
Singular integrals, Square functions, Layer potentials, Divergence form elliptic equations, Local Tb theorem
National Category
URN: urn:nbn:se:liu:diva-66901DOI: 10.1016/j.aim.2010.12.014ISI: 000287460000023OAI: diva2:405239
Available from: 2011-03-21 Created: 2011-03-21 Last updated: 2012-01-03Bibliographically approved

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Axelsson Rosén, Andreas
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Applied MathematicsThe Institute of Technology
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