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Boundary Behavior of p-Laplace Type Equations
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och tillämpad matematik.
2013 (engelsk)Doktoravhandling, med artikler (Annet vitenskapelig)
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.

sted, utgiver, år, opplag, sider
Uppsala: Acta Universitatis Upsaliensis, 2013. , s. 68
Serie
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology, ISSN 1651-6214 ; 1035
Emneord [en]
p-Laplace, Boundary Harnack inequality, A-harmonic, Ahlfors regularity, NTA-domains, Martin boundary, Reifenberg flat, Approximable by Lipschitz graphs, Subelliptic, Carleson estimate
HSV kategori
Forskningsprogram
Matematik
Identifikatorer
URN: urn:nbn:se:uu:diva-198008ISBN: 978-91-554-8645-7 (tryckt)OAI: oai:DiVA.org:uu-198008DiVA, id: diva2:615186
Disputas
2013-05-24, Polhemsalen, Lägerhyddsvägen 1, Uppsala, 10:15 (engelsk)
Opponent
Veileder
Tilgjengelig fra: 2013-05-03 Laget: 2013-04-08 Sist oppdatert: 2013-08-30
Delarbeid
1. Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
Åpne denne publikasjonen i ny fane eller vindu >>Estimates for Solutions to Equations of p-Laplace type in Ahlfors regular NTA-domains
2014 (engelsk)Inngår i: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 266, nr 9, s. 5955-6005Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-163517 (URN)10.1016/j.jfa.2014.02.027 (DOI)000334652000018 ()
Tilgjengelig fra: 2011-12-13 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
2. Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
Åpne denne publikasjonen i ny fane eller vindu >>Boundary estimates for solutions to operators of $p$-Laplace type with lower order terms
2011 (engelsk)Inngår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, nr 1, s. 264-291Artikkel i tidsskrift (Fagfellevurdert) Published
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-163370 (URN)10.1016/j.jde.2010.09.011 (DOI)
Tilgjengelig fra: 2011-12-12 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
3. Optimal doubling, Reifenberg flatness and operators of p-Laplace type
Åpne denne publikasjonen i ny fane eller vindu >>Optimal doubling, Reifenberg flatness and operators of p-Laplace type
2011 (engelsk)Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 17, s. 5943-5955Artikkel i tidsskrift (Fagfellevurdert) Published
Abstract [en]

In this paper we consider operators of p-Laplace type of the form ∇·A(x,∇u) = 0. ConcerningA we assume, for p ∈ (1,∞) fixed, an appropriate ellipticity type condition, H¨older continuityin x and that A(x, ) = ||p−1A(x, /||) whenever x ∈ Rn and ∈ Rn \ {0}. Let  ⊂ Rn be abounded domain, let D be a compact subset of . We say that ˆu = ˆup,D, is the A-capacitaryfunction for D in  if ˆu ≡ 1 on D, ˆu ≡ 0 on @ in the sense of W1,p0 () and ∇·A(x,∇ˆu) = 0 in \D in the weak sense. We extend ˆu to Rn \  by putting ˆu ≡ 0 on Rn \ . Then there existsa unique finite positive Borel measure ˆμ on Rn, with support in @, such thatZ hA(x,∇ˆu),∇i dx = −Z dˆμ whenever ∈ C∞0 (Rn \ D).In this paper we prove that if  is Reifenberg flat with vanishing constant, thenlimr→0infw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= limr→0supw∈∂ˆμ(B(w, r))ˆμ(B(w, r))= n−1,for every , 0 < ≤ 1. In particular, we prove that ˆμ is an asymptotically optimal doublingmeasure on @.

HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-163435 (URN)10.1016/j.na.2011.05.061 (DOI)
Tilgjengelig fra: 2012-11-30 Laget: 2011-12-12 Sist oppdatert: 2017-12-08bibliografisk kontrollert
4. Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
Åpne denne publikasjonen i ny fane eller vindu >>Boundary Estimates for Certain Degenerate and Singular Parabolic Equations
2016 (engelsk)Inngår i: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, nr 2, s. 381-424Artikkel i tidsskrift (Fagfellevurdert) 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.

Emneord
Degenerate and singular parabolic equations; Harnack estimates; boundary Harnack inequality; Carleson estimate
HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-186267 (URN)10.4171/JEMS/593 (DOI)000370249100005 ()
Tilgjengelig fra: 2013-02-12 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert
5. Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
Åpne denne publikasjonen i ny fane eller vindu >>Wolff-Potential Estimates and Doubling of Subelliptic p-harmonic measures
2013 (engelsk)Inngår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 85, s. 149-159Artikkel i tidsskrift (Fagfellevurdert) 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.

HSV kategori
Identifikatorer
urn:nbn:se:uu:diva-186268 (URN)10.1016/j.na.2013.02.023 (DOI)000318378700013 ()
Tilgjengelig fra: 2013-03-26 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert
6. On a one-phase free boundary problem
Åpne denne publikasjonen i ny fane eller vindu >>On a one-phase free boundary problem
2013 (engelsk)Inngår i: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, nr 1, s. 181-191Artikkel i tidsskrift (Annet vitenskapelig) 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.

Emneord
One-phase, free boundary, NTA, non-divergence, linear
HSV kategori
Forskningsprogram
Matematik
Identifikatorer
urn:nbn:se:uu:diva-186265 (URN)10.5186/aasfm.2013.3815 (DOI)000316239200009 ()
Tilgjengelig fra: 2012-11-30 Laget: 2012-11-28 Sist oppdatert: 2017-12-07bibliografisk kontrollert

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