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L2 Solvability of boundary value problems for divergence form parabolic equations with complex coefficients
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)Article in journal (Refereed) Submitted
Abstract [en]

We consider parabolic operators of the form $$\partial_t+\mathcal{L},\ \mathcal{L}=-\mbox{div}\, A(X,t)\nabla,$$ in$\mathbb R_+^{n+2}:=\{(X,t)=(x,x_{n+1},t)\in \mathbb R^{n}\times \mathbb R\times \mathbb R:\ x_{n+1}>0\}$, $n\geq 1$. We assume that $A$ is a $(n+1)\times (n+1)$-dimensional matrix which is bounded, measurable, uniformly elliptic and complex, and we assume, in addition, that the entries of A are independent of the spatial coordinate $x_{n+1}$ as well as of the time coordinate $t$. For such operators we prove that the boundedness and invertibility of the corresponding layer potential operators are stable on $L^2(\mathbb R^{n+1},\mathbb C)=L^2(\partial\mathbb R^{n+2}_+,\mathbb C)$ under complex, $L^\infty$ perturbations of the coefficient matrix. Subsequently, using this general result, we establish solvability of the Dirichlet, Neumann and Regularity problems for $\partial_t+\mathcal{L}$, by way of  layer potentials and with data in $L^2$,  assuming that the coefficient matrix is a small complex perturbation of either a constant matrix or of a real and symmetric matrix.

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URN: urn:nbn:se:uu:diva-268050OAI: oai:DiVA.org:uu-268050DiVA: diva2:875779
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2016-08-12Bibliographically approved

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Nyström, Kaj
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