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Boundary Estimates for Solutions to Parabolic Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation.

Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary.

Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary.

Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain.

Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric.

##### Place, publisher, year, edition, pages
Uppsala: Department of Mathematics , 2016. , 50 p.
##### Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 93
##### Keyword [en]
Uniformly parabolic equations, non-linear parabolic equations, linear growth, degenerate parabolic equations, fractional heat equations, complex coefficients, Lipschitz domain, NTA domain, boundary behaviour, boundary Harnack, parabolic measure, Riesz measure, Dirichlet to Neumann map, single layer potentials.
##### National Category
Mathematics Mathematical Analysis
Mathematics
##### Identifiers
ISBN: 978-91-506-2539-4OAI: oai:DiVA.org:uu-281451DiVA: diva2:914466
##### Public defence
2016-05-13, Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (English)
##### Supervisors
Available from: 2016-04-20 Created: 2016-03-24 Last updated: 2016-04-20
##### List of papers
1.
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2. Boundary estimates for solutions to linear degenerate parabolic equations
Open this publication in new window or tab >>Boundary estimates for solutions to linear degenerate parabolic equations
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.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204869 (URN)10.1016/j.jde.2015.04.028 (DOI)000363434300004 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2016-02-17Bibliographically approved
3. Extension properties and boundary estimates for a fractional heat operator
Open this publication in new window or tab >>Extension properties and boundary estimates for a fractional heat operator
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 140, 29-37 p.Article in journal (Refereed) Published
##### Abstract [en]

The square root of the heat operator $\sqrt{\partial_t-\Delta}$, can be realized as the Dirichlet to Neumann map of the heat extension of data on $\mathbb R^{n+1}$ to $\mathbb R^{n+2}_+$. In this note we obtain similar characterizations for general fractional powers of the heat operator, $(\partial_t-\Delta)^s$, $s\in (0,1)$. Using the characterizations we derive properties and boundary estimates for parabolic integro-differential equations from purely local arguments in the extension problem.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-266836 (URN)10.1016/j.na.2016.02.027 (DOI)000375484000003 ()
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2016-06-29Bibliographically approved
4. Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
Open this publication in new window or tab >>Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
2016 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835Article in journal (Refereed) Accepted
##### 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 an $(n+1)\times (n+1)$-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 $t$. We prove that the boundedness of associated single layer potentials, with data in $L^2$, can be reduced to two crucial estimates (Theorem \ref{th0}), one being a square function estimate involving the single layer potential. By establishing a local parabolic Tb-theorem for square functions we are then able to verify the two  crucial estimates in the case of real, symmetric operators (Theorem \ref{th2}). Our results are crucial when addressing the solvability of the classical Dirichlet, Neumann and Regularity problems for the operator $\partial_t+\mathcal{L}$ in $\mathbb R_+^{n+2}$, with $L^2$-data on $\mathbb R^{n+1}=\partial\mathbb R_+^{n+2}$, and by way of layer potentials.

Mathematics
##### Identifiers
urn:nbn:se:uu:diva-266835 (URN)
##### External cooperation:
Available from: 2015-11-11 Created: 2015-11-11 Last updated: 2016-08-29

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