Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables
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 result...
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Uppsala universitet, Analys och sannolikhetsteori
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ndltd-UPSALLA1-oai-DiVA.org-uu-2513252015-05-14T04:58:26ZStudies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex VariablesengPersson, HåkanUppsala universitet, Analys och sannolikhetsteoriUppsala2015uniformly parabolic equationsnon-linear parabolic equationslinear growthLipschitz domainNTA-domainRiesz measureboundary behaviorboundary Harnackdegenerate parabolicparabolic measureplurisubharmonic functionscontinuous boundaryhyperconvexitybounded exhaustion functionHölder for all exponentslog-lipschitzboundary regularityapproximationMergelyan type approximationplurisubharmonic functions on compactsJensen measuresmonotone convergenceplurisubharmonic extensionplurisubharmonic boundary valuesThis 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. Doctoral thesis, comprehensive summaryinfo:eu-repo/semantics/doctoralThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-251325urn:isbn:978-91-506-2458-8Uppsala Dissertations in Mathematics, 1401-2049 ; 89application/pdfinfo:eu-repo/semantics/openAccess |
collection |
NDLTD |
language |
English |
format |
Doctoral Thesis |
sources |
NDLTD |
topic |
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 |
spellingShingle |
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 Persson, Håkan Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
description |
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. |
author |
Persson, Håkan |
author_facet |
Persson, Håkan |
author_sort |
Persson, Håkan |
title |
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
title_short |
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
title_full |
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
title_fullStr |
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
title_full_unstemmed |
Studies of the Boundary Behaviour of Functions Related to Partial Differential Equations and Several Complex Variables |
title_sort |
studies of the boundary behaviour of functions related to partial differential equations and several complex variables |
publisher |
Uppsala universitet, Analys och sannolikhetsteori |
publishDate |
2015 |
url |
http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-251325 http://nbn-resolving.de/urn:isbn:978-91-506-2458-8 |
work_keys_str_mv |
AT perssonhakan studiesoftheboundarybehaviouroffunctionsrelatedtopartialdifferentialequationsandseveralcomplexvariables |
_version_ |
1716803717685051392 |