Change search

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
Mathematics
##### Identifiers
ISBN: 978-91-7459-287-0OAI: oai:DiVA.org:umu-47942DiVA: diva2:445532
##### Public defence
2011-10-28, Mit-huset, MA121, Umeå universitet, Umeå, 10:00
##### 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: 2015-12-10Bibliographically approved
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: 2014-09-15Bibliographically 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).

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: 2015-09-08Bibliographically 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.

##### Keyword
Boundary Harnack inequality, p-Laplace, A-harmonic function, infinity harmonic function, Aronsson type equation, quasicircle
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: 2015-03-16Bibliographically 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: 2015-03-16Bibliographically approved

#### Open Access in DiVA

##### File information
File name FULLTEXT01.pdfFile size 763 kBChecksum SHA-512
424e8f7b2881996a4f680ee427c3d14d0aac5937e35ea62280bfec2caed06270a000b122161e00bc65a19f9f288ea50e916036d089b2a35680b78d386df5cb33
Type fulltextMimetype application/pdf

#### Search in DiVA

##### By author/editor
Lundström, Niklas L.P.
##### By organisation
Department of Mathematics and Mathematical Statistics
##### On the subject
Mathematical Analysis