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p-harmonic functions near the boundary
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Place, publisher, year, edition, pages
Umeå: Umeå universitet, Institutionen för matematik och matematisk statistik , 2011. , 228 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 50
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-47942ISBN: 978-91-7459-287-0 (print)OAI: oai:DiVA.org:umu-47942DiVA: diva2:445532
Public defence
2011-10-28, Mit-huset, MA121, Umeå universitet, Umeå, 10:00
Opponent
Supervisors
Available from: 2011-10-07 Created: 2011-10-04 Last updated: 2011-10-04Bibliographically approved
List of papers
1. Estimates for p-harmonic functions vanishing on a flat
Open this publication in new window or tab >>Estimates for p-harmonic functions vanishing on a flat
2011 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, no 18, 6852-6860 p.Article in journal (Refereed) Published
Abstract [en]

We study p-harmonic functions in a domain ΩCRn near an m-dimensional plane (an m-flat) Λm, where 0≤mn−1. In particular, let u be a positive p-harmonic function, with n<p, vanishing on a portion of Λm, and suppose that β=(pn+m)/(p−1), with β=1 if p=. We prove, using certain barrier functions, that 

 u ≈ d (x.Λm)8   near Λm.

The lower bound holds also in the range nm<p. Moreover, uC0,β near Λm and β is the optimal Hölder exponent of u.

Keyword
Low dimension, Flat, Plane, Boundary Harnack inequality, p-Laplace, Infinity Laplace equation
National Category
Mathematical Analysis
Research subject
Mathematical Statistics; Mathematics
Identifiers
urn:nbn:se:umu:diva-47923 (URN)10.1016/j.na.2011.06.041 (DOI)000295714200002 ()
Note

Correction: Niklas L.P. Lundström, Corrigendum to “Estimates for p-harmonic functions vanishing on a flat” [Nonlinear Anal. 74(18) (2011) 6852–6860], Nonlinear Analysis: Theory, Methods & Applications, Volume 129, December 2015, Pages 371-372, ISSN 0362-546X, http://dx.doi.org/10.1016/j.na.2015.08.010

Available from: 2011-10-03 Created: 2011-10-03 Last updated: 2017-10-12
2. Decay of a p-harmonic measure in the plane
Open this publication in new window or tab >>Decay of a p-harmonic measure in the plane
2013 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 38, no 1, 351-366 p.Article in journal (Refereed) Published
Abstract [en]

We study the asymptotic behaviour of a p-harmonic measure w(p), p is an element of (1, infinity], in a domain Omega subset of R-2, subject to certain regularity constraints. Our main result is that w(p) (B (w, delta) boolean AND partial derivative Omega, w(0)) approximate to delta(q) as delta -> 0(+), where q = q(v,p) is given explicitly as a function of v and p. Here, v is related to properties of Omega near w. If p = infinity, this extends to some domains in R-n. By a result due to Hirata, our result implies that the p-Green function for p is an element of (1, 2) is not quasi-symmetric in plane C-1,C-1-domains.

Place, publisher, year, edition, pages
Suomalainen Tiedeakatemia, 2013
Keyword
harmonic measure, harmonic function, p-Laplace operator, generalized interior ball
National Category
Mathematics Mathematical Analysis
Research subject
Mathematics; Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-47922 (URN)10.5186/aasfm.2013.3808 (DOI)000316239200019 ()
Note

Originally published in thesis in manuscript form

Available from: 2011-10-03 Created: 2011-10-03 Last updated: 2017-12-08Bibliographically approved
3. On a two-phase free boundary condition for p-harmonic measures
Open this publication in new window or tab >>On a two-phase free boundary condition for p-harmonic measures
2009 (English)In: Manuscripta mathematica, ISSN 0025-2611, E-ISSN 1432-1785, Vol. 129, no 2, 231-249 p.Article in journal (Refereed) Published
Abstract [en]

Let Ωi⊂Rn,i∈{1,2} , be two (δ, r 0)-Reifenberg flat domains, for some 0<δ<δ^ and r 0 > 0, assume Ω1∩Ω2=∅ and that, for some w∈Rn and some 0 < r, w∈∂Ω1∩∂Ω2,∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . Let p, 1 < p < ∞, be given and let u i , i∈{1,2} , denote a non-negative p-harmonic function in Ω i , assume that u i , i∈{1,2}, is continuous in Ω¯i∩B(w,2r) and that u i = 0 on ∂Ωi∩B(w,2r) . Extend u i to B(w, 2r) by defining ui≡0 on B(w,2r)∖Ωi. Then there exists a unique finite positive Borel measure μ i , i∈{1,2} , on R n , with support in ∂Ωi∩B(w,2r) , such that if ϕ∈C∞0(B(w,2r)) , then∫Rn|∇ui|p−2⟨∇ui,∇ϕ⟩dx=−∫Rnϕdμi.Let Δ(w,2r)=∂Ω1∩B(w,2r)=∂Ω2∩B(w,2r) . The main result proved in this paper is the following. Assume that μ 2 is absolutely continuous with respect to μ 1 on Δ(w, 2r), d μ 2 = kd μ 1 for μ 1-almost every point in Δ(w, 2r) and that logk∈VMO(Δ(w,r),μ1) . Then there exists δ~=δ~(p,n)>0 , δ~<δ^ , such that if δ≤δ~ , then Δ(w, r/2) is Reifenberg flat with vanishing constant. Moreover, the special case p = 2, i.e., the linear case and the corresponding problem for harmonic measures, has previously been studied in Kenig and Toro (J Reine Angew Math 596:1–44, 2006).

Place, publisher, year, edition, pages
Springer, 2009
National Category
Mathematics Probability Theory and Statistics
Research subject
Mathematics; Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-31119 (URN)10.1007/s00229-009-0257-4 (DOI)000266010200005 ()
Available from: 2010-01-29 Created: 2010-01-29 Last updated: 2017-12-12Bibliographically approved
4. The Boundary Harnack Inequality for Solutions to Equations of Aronsson type in the Plane
Open this publication in new window or tab >>The Boundary Harnack Inequality for Solutions to Equations of Aronsson type in the Plane
2011 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 36, 261-278 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain \Omega \subset R2 and which are solutions to a general equation of p-Laplace type, 1 < p < \infty. We also establish the same type of result for solutions to the Aronsson type equation \nabla (F(x,\nabla u)) \cdot F\eta(x,\nabla u) = 0. Concerning \Omega we only assume that \partial\Omega is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

Place, publisher, year, edition, pages
Academia Scientiarum Fennica, 2011
Keyword
Boundary Harnack inequality, p-Laplace, A-harmonic function, infinity harmonic function, Aronsson type equation, quasicircle
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-40225 (URN)10.5186/aasfm.2011.3616 (DOI)
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2017-12-11Bibliographically approved
5. 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
Abstract [en]

In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain ΩRn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever xRn, ηRn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.

Keyword
Boundary Harnack inequality, p-harmonic function, A-harmonic function, (A, B)-harmonic function, Variable coefficients, Operators with lower order terms, Reifenberg flat domain, Martin boundary
National Category
Mathematics Probability Theory and Statistics
Research subject
Mathematical Statistics; Mathematics
Identifiers
urn:nbn:se:umu:diva-40224 (URN)10.1016/j.jde.2010.09.011 (DOI)000284919600013 ()
Available from: 2011-02-17 Created: 2011-02-17 Last updated: 2017-12-11Bibliographically approved

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