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Arnlind, J., Björn, A. & Björn, J. (2016). An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces. Nonlinear Analysis, 134, 70-104.
Open this publication in new window or tab >>An axiomatic approach to gradients with applications to Dirichlet and obstacle problems beyond function spaces
2016 (English)In: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 134, 70-104 p.Article in journal (Refereed) Published
Abstract [en]

We develop a framework for studying variational problems in Banach spaces with respect to gradient relations, which encompasses many of the notions of generalized gradients that appear in the literature. We stress the fact that our approach is not dependent on function spaces and therefore applies equally well to functions on metric spaces as to operator algebras. In particular, we consider analogues of Dirichlet and obstacle problems, as well as first eigenvalue problems, and formulate conditions for the existence of solutions and their uniqueness. Moreover, we investigate to what extent a lattice structure may be introduced on ( ordered) Banach spaces via a norm-minimizing variational problem. A multitude of examples is provided to illustrate the versatility of our approach. (C) 2015 Elsevier Ltd. All rights reserved.

Place, publisher, year, edition, pages
PERGAMON-ELSEVIER SCIENCE LTD, 2016
Keyword
Dirichlet problem; First eigenvalue; Generalized Sobolev space; Gradient relation; Lattice; Metric space; Noncommutative function; Obstacle problem; Operator-valued function; Partial order; Poincare set; Rayleigh quotient; Rellich-Kondrachov cone; Trace class ideal; Variational problem
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-126128 (URN)10.1016/j.na.2015.12.010 (DOI)000370489300004 ()
Note

Funding Agencies|Swedish Research Council

Available from: 2016-03-15 Created: 2016-03-15 Last updated: 2017-11-30
Björn, A., Björn, J. & Latvala, V. (2016). SOBOLEV SPACES, FINE GRADIENTS AND QUASICONTINUITY ON QUASIOPEN SETS. Annales Academiae Scientiarum Fennicae Mathematica, 41(2), 551-560.
Open this publication in new window or tab >>SOBOLEV SPACES, FINE GRADIENTS AND QUASICONTINUITY ON QUASIOPEN SETS
2016 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, E-ISSN 1798-2383, Vol. 41, no 2, 551-560 p.Article in journal (Refereed) Published
Abstract [en]

We study different definitions of Sobolev spaces on quasiopen sets in a complete metric space X equipped with a doubling measure supporting a p-Poincare inequality with 1 amp;lt; p amp;lt; infinity, and connect them to the Sobolev theory in R-n. In particular, we show that for quasiopen subsets of R-n the Newtonian functions, which are naturally defined in any metric space, coincide with the quasicontinuous representatives of the Sobolev functions studied by Kilpelainen and Maly in 1992.

Place, publisher, year, edition, pages
SUOMALAINEN TIEDEAKATEMIA, 2016
Keyword
Fine gradient; fine topology; metric space; minimal upper gradient; Newtonian space; quasicontinuous; quasiopen; Sobolev space
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-130151 (URN)10.5186/aasfm.2016.4130 (DOI)000378014700003 ()
Note

Funding Agencies|Swedish Research Council; Linkoping University; Institut Mittag-Leffler in the autumn

Available from: 2016-07-12 Created: 2016-07-11 Last updated: 2017-11-28
Björn, J. (2016). The Dirichlet problem and boundary regularity for nonlinear parabolic equations. In: : . Paper presented at 27th Nordic Congress of Mathematicians, Stockholm, 16-20 March 2016. .
Open this publication in new window or tab >>The Dirichlet problem and boundary regularity for nonlinear parabolic equations
2016 (English)Conference paper, Oral presentation only (Other academic)
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-126767 (URN)
Conference
27th Nordic Congress of Mathematicians, Stockholm, 16-20 March 2016
Available from: 2016-04-04 Created: 2016-04-04 Last updated: 2016-04-21
Björn, A., Björn, J. & Shanmugalingam, N. (2016). The Mazurkiewicz Distance and Sets that are Finitely Connected at the Boundary. Journal of Geometric Analysis, 26(2), 873-897.
Open this publication in new window or tab >>The Mazurkiewicz Distance and Sets that are Finitely Connected at the Boundary
2016 (English)In: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 26, no 2, 873-897 p.Article in journal (Refereed) Published
Abstract [en]

We study local connectedness, local accessibility and finite connectedness at the boundary, in relation to the compactness of the Mazurkiewicz completion of a bounded domain in a metric space. For countably connected planar domains we obtain a complete characterization. It is also shown exactly which parts of this characterization fail in higher dimensions and in metric spaces.

Place, publisher, year, edition, pages
SPRINGER, 2016
Keyword
Compactness; Countably connected planar domain; Finitely connected at the boundary; Locally accessible; Locally connected; Mazurkiewicz boundary; Metric space
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-130310 (URN)10.1007/s12220-015-9575-9 (DOI)000378524800007 ()
Note

Funding Agencies|Swedish Research Council; Swedish Fulbright Commission; Charles Phelps Taft Research Center at the University of Cincinnati; Taft Research Center of the University of Cincinnati; Simons Foundation [200474]; NSF [DMS-1200915]

Available from: 2016-07-31 Created: 2016-07-28 Last updated: 2017-11-28
Björn, A., Björn, J., Gianazza, U. & Parviainen, M. (2015). Boundary regularity for degenerate and singular parabolic equations. Calculus of Variations and Partial Differential Equations, 52(3-4), 797-827.
Open this publication in new window or tab >>Boundary regularity for degenerate and singular parabolic equations
2015 (English)In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 52, no 3-4, 797-827 p.Article in journal (Refereed) Published
Abstract [en]

We characterise regular boundary points of the parabolic p-Laplacian in terms of a family of barriers, both when p greater than 2 and 1 less than p less than 2. By constructing suitable families of such barriers, we give some simple geometric conditions that ensure the regularity of boundary points.

Place, publisher, year, edition, pages
Springer Verlag (Germany), 2015
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-117660 (URN)10.1007/s00526-014-0734-9 (DOI)000352199600017 ()
Note

Funding Agencies|Swedish Research Council; Academy of Finland

Available from: 2015-05-12 Created: 2015-05-06 Last updated: 2017-12-04
Björn, J. (2015). Boundary regularity for quasiminimizers. In: : . Paper presented at Workshop on Analysis and Geometry in Metric Spaces, ICMAT, Madrid, Spain, 1-5 June 2015. .
Open this publication in new window or tab >>Boundary regularity for quasiminimizers
2015 (English)Conference paper, Oral presentation with published abstract (Other academic)
Abstract [en]

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123943 (URN)
Conference
Workshop on Analysis and Geometry in Metric Spaces, ICMAT, Madrid, Spain, 1-5 June 2015
Available from: 2016-01-14 Created: 2016-01-14 Last updated: 2016-05-04Bibliographically approved
Björn, J. (2015). Boundary regularity for quasiminimizers. In: : . Paper presented at Workshop on Geometrical Analysis, dedicated to the 60th birthday of Jan Maly, Prague, Czech Republic, 18-20 September 2015. .
Open this publication in new window or tab >>Boundary regularity for quasiminimizers
2015 (English)Conference paper, Oral presentation with published abstract (Other academic)
Abstract [en]

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-123944 (URN)
Conference
Workshop on Geometrical Analysis, dedicated to the 60th birthday of Jan Maly, Prague, Czech Republic, 18-20 September 2015
Available from: 2016-01-14 Created: 2016-01-14 Last updated: 2016-05-04Bibliographically approved
Björn, A. & Björn, J. (2015). Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology. Revista matemática iberoamericana, 31(1), 161-214.
Open this publication in new window or tab >>Obstacle and Dirichlet problems on arbitrary nonopen sets in metric spaces, and fine topology
2015 (English)In: Revista matemática iberoamericana, ISSN 0213-2230, E-ISSN 2235-0616, Vol. 31, no 1, 161-214 p.Article in journal (Refereed) Published
Abstract [en]

We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain the Adams criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Many of the results are new even for open E (apart from those which are trivial in this case) and also on R-n.

Place, publisher, year, edition, pages
Universidad Autonoma de Madrid, Departamento de Matematicas / European Mathematical Society, 2015
Keyword
Adams criterion; Dirichlet problem; doubling measure; fine potential theory; Friedrichs inequality; metric space; minimal upper gradient; nonlinear; obstacle problem; p-harmonic; Poincare inequality; potential theory; upper gradient
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-117820 (URN)10.4171/RMI/830 (DOI)000352568500007 ()
Note

Funding Agencies|Swedish Research Council

Available from: 2015-05-11 Created: 2015-05-08 Last updated: 2017-12-04
Björn, A., Björn, J. & Shanmugalingam, N. (2015). The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities. Journal of Differential Equations, 259(7), 3078-3114.
Open this publication in new window or tab >>The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary, and new capacities
2015 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 7, 3078-3114 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we develop the Perron method for solving the Dirichlet problem for the analog of the p-Laplacian, i.e. for p-harmonic functions, with Mazurkiewicz boundary values. The setting considered here is that of metric spaces, where the boundary of the domain in question is replaced with the Mazurkiewicz boundary. Resolutivity for Sobolev and continuous functions, as well as invariance results for perturbations on small sets, are obtained. We use these results to improve the known resolutivity and invariance results for functions on the standard (metric) boundary. We also illustrate the results of this paper by discussing several examples. (C) 2015 Elsevier Inc. All rights reserved.

Place, publisher, year, edition, pages
Elsevier, 2015
Keyword
Dirichlet problem; Finite connectivity at the boundary; Mazurkiewicz distance; Metric space; p-harmonic function; Perron method
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-120434 (URN)10.1016/j.jde.2015.04.014 (DOI)000357903500018 ()
Note

Funding Agencies|Swedish Research Council; Swedish Fulbright Commission; Charles Phelps Taft Research Center at the University of Cincinnati; Taft Research Center; Simons Foundation [200474]; NSF grant [DMS-1200915]

Available from: 2015-08-12 Created: 2015-08-11 Last updated: 2017-12-04
Björn, A., Björn, J. & Latvala, V. (2015). The Weak Cartan Property for the p-fine Topology on Metric Spaces. Indiana University Mathematics Journal, 64(3), 915-941.
Open this publication in new window or tab >>The Weak Cartan Property for the p-fine Topology on Metric Spaces
2015 (English)In: Indiana University Mathematics Journal, ISSN 0022-2518, E-ISSN 1943-5258, Vol. 64, no 3, 915-941 p.Article in journal (Refereed) Published
Abstract [en]

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 less than p less than infinity. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all p-superharmonic functions continuous. Our p-harmonic and superharmonic functions are defined by means of scalar-valued upper gradients, and do not rely on a vector-valued differentiable structure.

Place, publisher, year, edition, pages
INDIANA UNIV MATH JOURNAL, 2015
Keyword
Capacity; coarsest topology; doubling; fine topology finely continuous; metric space; p-harmonic; Poincare inequality; quasi-continuous; superharmonic; thick; thin; weak Cartan property; Wiener criterion
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-120665 (URN)000358271700009 ()
Note

Funding Agencies|Swedish Research Council; Scandinavian Research Network Analysis and Application,; Linkopings universitet

Available from: 2015-08-20 Created: 2015-08-20 Last updated: 2017-12-04
Organisations
Identifiers
ORCID iD: ORCID iD iconorcid.org/0000-0002-1238-6751

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