Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Boundary Behavior of p-Laplace Type Equations
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six scientific papers, an introduction and a summary. All six papers concern the boundary behavior of non-negative solutions to partial differential equations.

Paper I concerns solutions to certain p-Laplace type operators with variable coefficients. Suppose that u is a non-negative solution that vanishes on a part Γ of an Ahlfors regular NTA-domain. We prove among other things that the gradient Du of u has non-tangential limits almost everywhere on the boundary piece Γ, and that log|Du| is a BMO function on the boundary.  Furthermore, for Ahlfors regular NTA-domains that are uniformly (N,δ,r0)-approximable by Lipschitz graph domains we prove a boundary Harnack inequality provided that δ is small enough. 

Paper II concerns solutions to a p-Laplace type operator with lower order terms in δ-Reifenberg flat domains. We prove that the ratio of two non-negative solutions vanishing on a part of the boundary is Hölder continuous provided that δ is small enough. Furthermore we solve the Martin boundary problem provided δ is small enough.

In Paper III we prove that the boundary type Riesz measure associated to an A-capacitary function in a Reifenberg flat domain with vanishing constant is asymptotically optimal doubling.

Paper IV concerns the boundary behavior of solutions to certain parabolic equations of p-Laplace type in Lipschitz cylinders. Among other things, we prove an intrinsic Carleson type estimate for the degenerate case and a weak intrinsic Carleson type estimate in the singular supercritical case.

In Paper V we are concerned with equations of p-Laplace type structured on Hörmander vector fields. We prove that the boundary type Riesz measure associated to a non-negative solution that vanishes on a part Γ of an X-NTA-domain, is doubling on Γ.

Paper VI concerns a one-phase free boundary problem for linear elliptic equations of non-divergence type. Assume that we know that the positivity set is an NTA-domain and that the free boundary is a graph. Furthermore assume that our solution is monotone in the graph direction and that the coefficients of the equation are constant in the graph direction. We prove that the graph giving the free boundary is Lipschitz continuous.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2013. , 68 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1035
Keyword [en]
p-Laplace, Boundary Harnack inequality, A-harmonic, Ahlfors regularity, NTA-domains, Martin boundary, Reifenberg flat, Approximable by Lipschitz graphs, Subelliptic, Carleson estimate
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-198008ISBN: 978-91-554-8645-7 (print)OAI: oai:DiVA.org:uu-198008DiVA: diva2:615186
Public defence
2013-05-24, Polhemsalen, Lägerhyddsvägen 1, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2013-05-03 Created: 2013-04-08 Last updated: 2013-08-30
List of papers
1. Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
Open this publication in new window or tab >>Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, no 9, 5955-6005 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-163517 (URN)10.1016/j.jfa.2014.02.027 (DOI)000334652000018 ()
Available from: 2011-12-13 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
2. Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
Open this publication in new window or tab >>Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
2011 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, no 1, 264-291 p.Article in journal (Refereed) Published
National Category
Mathematics
Identifiers
urn:nbn:se:uu:diva-163370 (URN)10.1016/j.jde.2010.09.011 (DOI)
Available from: 2011-12-12 Created: 2011-12-12 Last updated: 2017-12-08Bibliographically approved
3.
The record could not be found. The reason may be that the record is no longer available or you may have typed in a wrong id in the address field.
4. Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
Open this publication in new window or tab >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
2016 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 2, 381-424 p.Article in journal (Refereed) Published
Abstract [en]

We study the boundary behavior of non-negative solutions to a class of degenerate/singular parabolic equations, whose prototype is the parabolic p-Laplace equation. Assuming that such solutions continuously vanish on some distinguished part of the lateral part S-T of a Lipschitz cylinder, we prove Carleson-type estimates, and deduce some consequences under additional assumptions on the equation or the domain. We then prove analogous estimates for non-negative solutions to a class of degenerate/singular parabolic equations of porous medium type.

Keyword
Degenerate and singular parabolic equations; Harnack estimates; boundary Harnack inequality; Carleson estimate
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-186267 (URN)10.4171/JEMS/593 (DOI)000370249100005 ()
Available from: 2013-02-12 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
5. Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
Open this publication in new window or tab >>Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
2013 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, 149-159 p.Article in journal (Refereed) Published
Abstract [en]

Let be a system of C vector fields in Rn satisfying Hörmander’s finite rank condition and let Ω be a non-tangentially accessible domain with respect to the Carnot–Carathéodory distance d induced by X. We prove the doubling property of certain boundary measures associated to non-negative solutions, which vanish on a portion of Ω, to the equation

Given p, 1<p<, fixed, we impose conditions on the function A=(A1,…,Am):Rn×RmRm, which imply that the equation is a quasi-linear partial differential equation of p-Laplace type structured on vector fields satisfying the classical Hörmander condition. In the case p=2 and for linear equations, our result coincides with the doubling property of associated elliptic measures. To prove our result we establish, and this is of independent interest, a Wolff potential estimate for subelliptic equations of p-Laplace type.

National Category
Mathematical Analysis
Identifiers
urn:nbn:se:uu:diva-186268 (URN)10.1016/j.na.2013.02.023 (DOI)000318378700013 ()
Available from: 2013-03-26 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved
6. On a one-phase free boundary problem
Open this publication in new window or tab >>On a one-phase free boundary problem
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, 181-191 p.Article in journal (Other academic) Published
Abstract [en]

In this paper we extend a result regarding the free boundary regularity in a one-phaseproblem, by De Silva and Jerison [DJ], to non-divergence linear equations of second order.Roughly speaking we prove that the free boundary is given by a Lipschitz graph.

Keyword
One-phase, free boundary, NTA, non-divergence, linear
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
urn:nbn:se:uu:diva-186265 (URN)10.5186/aasfm.2013.3815 (DOI)000316239200009 ()
Available from: 2012-11-30 Created: 2012-11-28 Last updated: 2017-12-07Bibliographically approved

Open Access in DiVA

fulltext(1450 kB)906 downloads
File information
File name FULLTEXT01.pdfFile size 1450 kBChecksum SHA-512
de96df62e5668794454582aaf90f1e2566d84880d0434fa6b36e77e48c45a1d7ecd361905426821f75120136290c7570d4e0744e8d4c731c6ba429bf1762d023
Type fulltextMimetype application/pdf
Buy this publication >>

Authority records BETA

Avelin, Benny

Search in DiVA

By author/editor
Avelin, Benny
By organisation
Analysis and Applied Mathematics
Mathematical Analysis

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

isbn
urn-nbn

Altmetric score

isbn
urn-nbn
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf