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Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables.

Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property.

Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property.

In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given.

In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain.

Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set.

Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.

##### Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2015. , p. 52
##### Series
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 89
##### Keyword [en]
uniformly parabolic equations, non-linear parabolic equations, linear growth, Lipschitz domain, NTA-domain, Riesz measure, boundary behavior, boundary Harnack, degenerate parabolic, parabolic measure, plurisubharmonic functions, continuous boundary, hyperconvexity, bounded exhaustion function, Hölder for all exponents, log-lipschitz, boundary regularity, approximation, Mergelyan type approximation, plurisubharmonic functions on compacts, Jensen measures, monotone convergence, plurisubharmonic extension, plurisubharmonic boundary values
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
ISBN: 978-91-506-2458-8 (print)OAI: oai:DiVA.org:uu-251325DiVA, id: diva2:805443
##### Public defence
2015-06-05, Polhemssalen, Ångströmslaboratoriet, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
##### Supervisors
Available from: 2015-05-13 Created: 2015-04-15 Last updated: 2015-05-13
##### List of papers
1. Boundary estimates for non-negative solutions to non-linear parabolic equations
Open this publication in new window or tab >>Boundary estimates for non-negative solutions to non-linear parabolic equations
2015 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 54, no 1, p. 847-879Article in journal (Refereed) Published
##### Abstract [en]

This paper is mainly devoted to  the boundary behavior of non-negative solutions to the equation$\H u =\partial_tu-\nabla\cdot \operatorname{A}(x,t,\nabla u) = 0$in domains of the form $\Omega_T=\Omega\times (0,T)$ where $\Omega\subset\mathbb R^n$ is a bounded non-tangentially accessible (NTA) domain and $T>0$. The assumptions we impose on$A$ imply that $H$ is a non-linear parabolic operator with linear growth. Our main results include a backward Harnackinequality, and the H\"older continuity  up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary. Furthermore, to each such solution one can associate a natural Riesz measure supported on the lateral boundary and one of our main result is a proof of the doubling property for this measure. Our results generalize,  to the setting of non-linear equations with linear growth, previous results concerningthe boundary behaviour, in Lipschitz cylinders and time-independent NTA-cylinders, established for  non-negative solutions to equations of the type $\partial_tu-\nabla\cdot (\operatorname{A}(x,t)\nabla u)=0$, where $A$ is a measurable, bounded and uniformly positive definite matrix-valued function. In the latter case the measure referred to above is essentially the caloric or parabolic measure associated to  the operator and related to Green's function. At the end of the paper we also remark that our arguments are general enough to allow us to generalize parts of our results to general fully non-linear parabolic partial differential equations of second order.

Mathematics
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-204871 (URN)10.1007/s00526-014-0808-8 (DOI)000359941200033 ()
Available from: 2013-08-12 Created: 2013-08-12 Last updated: 2017-12-06Bibliographically approved
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, p. 3577-3614Article 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: 2017-12-06Bibliographically approved
3. A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
Open this publication in new window or tab >>A note on the hyperconvexity of pseudoconvex domains beyond Lipschitz regularity
2015 (English)In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, no 3, p. 531-545Article in journal (Refereed) Published
##### Abstract [en]

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 184 1987) beyond Lipschitz regularity.

##### Keyword
plurisubharmonic functions, continuous boundary, hyperconvexity, bounded exhaustion function, Hölder for all exponents, log-lipschitz, boundary regularity, Reinhardt domains.
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-251330 (URN)10.1007/s11118-015-9486-1 (DOI)000365769100010 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
4. Approximation of plurisubharmonic functions
Open this publication in new window or tab >>Approximation of plurisubharmonic functions
2016 (English)In: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, no 1, p. 23-28Article in journal (Refereed) Published
##### Abstract [en]

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

##### Keyword
plurisubharmonic functions, approximation, continuous boundary, boundary regularity, Mergelyan type approximation
##### National Category
Mathematical Analysis
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-251324 (URN)10.1080/17476933.2015.1053473 (DOI)000365643500003 ()
Available from: 2015-04-15 Created: 2015-04-15 Last updated: 2017-12-04Bibliographically approved
5. Plurisubharmonic functions on compact sets
Open this publication in new window or tab >>Plurisubharmonic functions on compact sets
2012 (English)In: Annales Polonici Mathematici, ISSN 0066-2216, E-ISSN 1730-6272, Vol. 106, p. 133-144Article in journal (Refereed) Published
##### Abstract [en]

Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in C-n. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.

##### Keyword
plurisubharmonic functions on compacts, Jensen measures, monotone convergence
Natural Sciences
##### Identifiers
urn:nbn:se:uu:diva-189931 (URN)10.4064/ap106-0-11 (DOI)000311525700011 ()
Available from: 2013-01-04 Created: 2013-01-04 Last updated: 2017-12-06Bibliographically approved
6. Plurisubharmonic approximation and boundary values of plurisubharmonic functions
Open this publication in new window or tab >>Plurisubharmonic approximation and boundary values of plurisubharmonic functions
2014 (English)In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 413, no 2, p. 700-714Article in journal (Refereed) Published
##### Abstract [en]

We study the problem of approximating plurisubharmonic functions on a bounded domain Omega by continuous plurisubharmonic functions defined on neighborhoods of (Omega) over bar. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set (Omega) over bar in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.

##### Keyword
Plurisubharmonic functions on compacts, Jensen measures, Approximation, Plurisubharmonic extension, Plurisubharmonic boundary values
Mathematics
##### Identifiers
urn:nbn:se:uu:diva-220970 (URN)10.1016/j.jmaa.2013.12.041 (DOI)000331344600014 ()
Available from: 2014-03-26 Created: 2014-03-24 Last updated: 2017-12-05Bibliographically approved

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Cite
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