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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.
2017 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 262, no 3, 2808-2939 p.Article in journal (Refereed) Published
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.

Place, publisher, year, edition, pages
2017. Vol. 262, no 3, 2808-2939 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-268050DOI: 10.1016/j.jde.2016.11.011ISI: 000392463200026OAI: oai:DiVA.org:uu-268050DiVA: diva2:875779
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved

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Publisher's full textarXiv:1603.02823

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Nyström, Kaj

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