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Boundary estimates for solutions to linear degenerate parabolic equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 8, 3577-3614 p.Article in journal (Refereed) Published
Abstract [en]

Let $\Omega\subset\mathbb R^n$ be a bounded NTA-domain and let $\Omega_T=\Omega\times (0,T)$ for some $T>0$.  We study the boundary behaviour of non-negativesolutions to the equation\[Hu =\partial_tu-\partial_{x_i}(a_{ij}(x,t)\partial_{x_j}u) = 0, \ (x,t)\in \Omega_T.\]We assume that $A(x,t)=\{a_{ij}(x,t)\}$ is measurable, real, symmetric and that\begin{equation*}\beta^{-1}\lambda(x)|\xi|^2\leq \sum_{i,j=1}^na_{ij}(x,t)\xi_i\xi_j\leq\beta\lambda(x)|\xi|^2\mbox{ for all }(x,t)\in\mathbb R^{n+1},\ \xi\in\mathbb R^{n},\end{equation*}for some constant $\beta\geq 1$ and for some non-negative and real-valued function $\lambda=\lambda(x)$belonging to the Muckenhoupt class $A_{1+2/n}(\mathbb R^n)$.Our main results includethe doubling property of the associated parabolic measure andthe H\"older continuity  up to the boundary of quotients of non-negative solutionswhich vanish continuously on a portion of the boundary. Our resultsgeneralize previous results of Fabes, Kenig, Jerison, Serapioni, see \cite{FKS}, \cite{FJK}, \cite{FJK1}, to a parabolic setting.

Place, publisher, year, edition, pages
2015. Vol. 259, no 8, 3577-3614 p.
National Category
Mathematics
Research subject
Mathematics
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URN: urn:nbn:se:uu:diva-204869DOI: 10.1016/j.jde.2015.04.028ISI: 000363434300004OAI: oai:DiVA.org:uu-204869DiVA: diva2:639981
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved

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Nyström, KajPersson, HåkanSande, Olow

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