Nonlinear Potential Theory

Our research group is mainly interested in nonlinear potential theory associated with p-harmonic functions and quasiminimizers in Euclidean and metric spaces. We are also interested in first-order Sobolev spaces, in particular the so called Newtonian spaces on metric spaces.

We organized the conference: Nonlinear problems for $\Delta_p$ and $\Delta$, 10-14 August 2009 in Linköping.

Members of the group

Tomasz Adamowicz
Anders Björn
Jana Björn
Zohra Farnana
Daniel Hansevi
Lukáš Malý
Tomas Sjödin

Collaboration

People with whom we collaborate or have close contacts with include the following people.
Stephen Gardiner at University College Dublin
Juha Kinnunen at Helsinki University of Technology
Riikka Korte, Niko Marola and Olli Martio at Helsinki University
Visa Latvala at Joensuu University
Mikko Parviainen and Xiao Zhong at Jyväskylä University
Nageswari Shanmugalingam at University of Cincinnati
Jan Malý at Charles University, Prague
Stephen Buckley at University of Maynooth

Publications

The list below contains the mathematical publications of the members of the group.
The list is sorted in reverse chronological order.

Books

A. Björn and J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Mathematics 17, European Mathematical Society, Zurich, 2011, 415 pp, ISBN 978-3-03719-099-9. Distributed by EMS and AMS. The profit from this book is donated to Barndiabetesfonden (The Swedish Child Diabetes Fund).
A. Asratian, A. Björn and B. O. Turesson, Diskret Matematik, 12th ed., Matematiska institutionen, Linköpings universitet, 2009, 232 pp. (Swedish).

Ph.D. theses

Z. Farnana, The Double Obstacle Problem on Metric Spaces, Ph.D. Dissertation, Linköping Studies in Science and Technology. Dissertations No. 1283, Linköping, 2009, 94 pp.
T. Sjödin, Topics in Potential Theory: Quadrature Domains, Balayage and Harmonic Measure, Doctoral Thesis in Mathematics, TRITA-MAT-05-MA-05, Royal Institute of Technology, Stockholm, 2005, 232 pp.
J. Björn, Interior Regularity and Boundary Behaviour of Solutions to Second Order Elliptic Equations, Ph.D. Dissertation, Linköping Studies in Science and Technology. Dissertations No. 446, Linköping, 1996, 92 pp.
A. Björn, Removable Singularities for Hardy Spaces of Analytic Functions, Ph.D. Dissertation, Linköping Studies in Science and Technology. Dissertations No. 365, Linköping, 1994, 74 pp.

Licentiate theses

Z. Farnana, The Double Obstacle Problem on Metric Spaces, Licentiate thesis, Linköping Studies in Science and Technology. Theses. No. 1342, Linköping, 2008, 52 pp.
For Master's and Bachelor's theses see at the bottom.

Preprints

T. Adamowicz, A. Björn, J. Björn and N. Shanmugalingam, Prime ends for domains in metric spaces, Preprint, 2012. arXiv

Refereed articles

To appear
L. Malý, Calderón-type theorems for operators with non-standard endpoint behavior on Lorentz spaces, to appear in Math. Nachr.
H. Shahgholian and T. Sjödin, Harmonic balls and the two-phase Schwarz function, to appear in Complex Var. Elliptic Equ. Journal arXiv

2012
S.J. Gardiner and T. Sjödin, Two-phase quadrature domains, J. Anal. Math. 116 (2012), 335--354. Preprint

2011
A. Björn and H. Riesel Table errata 2 to "Factors of generalized Fermat numbers", Math. Comp. 80 (2011), 1865--1866.
A. Björn and J. Björn, Power-type quasiminimizers, Ann. Acad. Sci. Fenn. Math. 36 (2011), 301--319.
Z. Farnana, Pointwise regularity for solutions of double obstacle problems on metric spaces, Math. Scand. 109 (2011), 185--200.

2010
A. Björn, p-harmonic functions with boundary data having jump discontinuities and Baernstein's problem, J. Differential Equations 249 (2010), 1--36.
A. Björn, Cluster sets for Sobolev functions and quasiminimizers, J. Anal. Math. 112 (2010), 49--77.
A. Björn, J. Björn and N. Marola, BMO, integrability, Harnack and Caccioppoli inequalities for quasiminimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 1489--1505.
A. Björn, J. Björn and M. Parviainen, Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces, Rev. Mat. Iberoam. 26 (2010), 147--174.
J. Björn, Orlicz--Poincaré inequalities, maximal functions and $A_\Phi$-conditions, Proc. Roy. Soc. Edinburgh Sect. A 140 (2010), 31--48.
M. Eleuteri, Z. Farnana, O. E. Kansanen and R. Korte, Stability of solutions of the double obstacle problem on metric spaces, J. Anal. 18 (2010), 145--160. Journal
Z. Farnana, Continuous dependence on obstacles for the double obstacle problem on metric spaces, Nonlinear Anal. 73 (2010), 2819--2830.
Z. Farnana, Convergence results for obstacle problems on metric spaces, J. Math. Anal. Appl. 371 (2010), 436--446.

2009
A. Björn, J. Björn, T. Mäkäläinen and M. Parviainen, Nonlinear balayage on metric spaces, Nonlinear Anal. 71 (2009), 2153--2171.
A. Björn and O. Martio, Pasting lemmas and characterizations of boundary regularity for quasiminimizers, Results Math. 55 (2009), 265--279.
J. Björn, Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations, Calc. Var. Partial Differential Equations 35 (2009), 481--496.
Z. Farnana, The double obstacle problem on metric spaces, Ann. Acad. Sci. Fenn. Math. 34 (2009), 261--277.

2008
A. Björn, A regularity classification of boundary points for p-harmonic functions and quasiminimizers, J. Math. Anal. Appl. 338 (2008), 39--47.
A. Björn, J. Björn and N. Shanmugalingam, Quasicontinuity of Newton-Sobolev functions and density of Lipschitz functions on metric spaces, Houston J. Math. 34 (2008), 1197--1211.
J. Björn, Fine continuity on metric spaces, Manuscripta Math. 125 (2008), 369--381.
S. J. Gardiner and T. Sjödin, Convexity and the exterior inverse problem of potential theory, Proc. Amer. Math. Soc. 136 (2008), 1699--1703.

2007
A. Björn, Weak barriers in nonlinear potential theory, Potential Anal. 27 (2007), 381--387.
A. Björn and J. Björn, Approximations by regular sets and Wiener solutions in metric spaces, Comment. Math. Univ. Carolin. 48 (2007), 343--355.
A. Björn, J. Björn and N. Shanmugalingam Sobolev extensions of Hölder continuous and characteristic functions on metric spaces, Canadian J. Math. 59 (2007), 1135--1153.
J. Björn and N. Shanmugalingam, Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces, J. Math. Anal. Appl. 332 (2007), 190--208.
S. J. Gardiner and T. Sjödin, Quadrature domains for harmonic functions, Bull. Lond. Math. Soc. 39 (2007), 586--590.
T. Sjödin, On the structure of partial balayage, Nonlinear Anal. 67 (2007), 94--102.
T. Sjödin, Approximation in the cone of positive harmonic functions, Potential Anal. 27 (2007), 271--280.

2006
A. Björn, Removable singularities for bounded p-harmonic and quasi(super)harmonic functions, Ann. Acad. Sci. Fenn. Math. 31 (2006), 71--95.
A. Björn, Removable singularities in weighted Bergman spaces, Czechoslovak Math. J. 56 (2006), 179--227.
A. Björn, A weak Kellogg property for quasiminimizers, Comment. Math. Helv. 81 (2006), 809--825.
A. Björn and J. Björn, Boundary regularity for p-harmonic functions and solutions of the obstacle problem, J. Math. Soc. Japan 58 (2006), 1211--1232.
A. Björn, J. Björn and N. Shanmugalingam, A problem of Baernstein on the equality of the p-harmonic measure of a set and its closure, Proc. Amer. Math. Soc. 134 (2006), 509--519.
A. Björn and N. Marola, Moser iteration for (quasi)minimizers on metric spaces, Manuscripta Math. 121 (2006), 339--366.
J. Björn, S. Buckley and S. Keith, Admissible measures in one dimension, Proc. Amer. Math. Soc. 134 (2006), 703--705.
T. Sjödin, Mother bodies of algebraic domains in the complex plane, Complex Var. Elliptic Equ. 51 (2006), 357--369.

2005
A. Björn, Characterizations of p-superharmonic functions on metric spaces, Studia Math. 169 (2005), 45--62.
A. Björn and H. Riesel, Table errata on "Factors of generalized Fermat numbers", Math. Comp. 74 (2005), 2099.
J. Björn and J. Onninen, Orlicz capacities and Hausdorff measures on metric spaces, Math. Z. 251 (2005), 131--146.

2003
A. Björn, Removable singularities for analytic functions in BMO and locally Lipschitz spaces, Math. Z. 244 (2003), 805--835.
A. Björn, J. Björn and N. Shanmugalingam, The Dirichlet problem for p-harmonic functions on metric spaces, J. Reine Angew. Math. 556 (2003), 173--203.
A. Björn, J. Björn and N. Shanmugalingam, The Perron method for p-harmonic functions, J. Differential Equations 195 (2003), 398--429.

2002
N. Arcozzi and A. Björn, Dominating sets for analytic and harmonic functions and completeness of weighted Bergman spaces, Math. Proc. Roy. Irish Acad. 102A (2002), 175--192.
A. Björn, Properties of removable singularities for Hardy spaces of analytic functions, J. Lond. Math. Soc. 66 (2002), 651--670.
J. Björn, Boundary continuity for quasiminimizers on metric spaces, Illinois J. Math. 46 (2002), 383--403.
J. Björn, Mappings with dilatation in Orlicz spaces, Collectanea Math. 53 (2002), 303--311.

2001
A. Björn, Removable singularities for H ^p spaces of analytic functions, 0<p<1, Ann. Acad. Sci. Fenn. Math. 26 (2001), 155--174.
J. Björn, Boundedness and differentiability for nonlinear elliptic systems, Trans. Amer. Math. Soc. 353 (2001), 4545--4565.
J. Björn, Poincar\'e inequalities for powers and products of admissible weights, Ann. Acad. Sci. Fenn. Math. 26 (2001), 175--188.
J. Björn, P. MacManus and N. Shanmugalingam, Fat sets and pointwise boundary estimates for p-harmonic functions in metric spaces, J. Anal. Math. 85 (2001), 339--369.

2000
J. Björn, L^q-differentials for weighted Sobolev spaces, Michigan Math. J. 47 (2000), 151--161.
J. Björn and V. Maz'ya, Capacitary estimates for solutions of the Dirichlet problem for second order elliptic equations in divergence form, Potential Anal. 12 (2000), 81--113.

1998
A. Björn Removable singularities for Hardy spaces, Complex Variables Theory Appl. 35 (1998), 1--25.
A. Björn, Removable singularities on rectifiable curves for Hardy spaces of analytic functions. Math. Scand. 83 (1998), 87--102.
A. Björn and Riesel, H., Factors of generalized Fermat numbers, Math. Comp. 67 (1998), 441--446 + 49 pp. of tables on micro-fiche.

1997
J. Björn, Regularity at infinity for a mixed problem for degenerate elliptic operators in a half-cylinder, Math. Scand. 81 (1997), 101--126.

1994
J. Jezková [Björn], Boundedness and pointwise differentiability of weak solutions to quasi-linear elliptic differential equations and variational inequalities, Comment. Math. Univ. Carolin. 35 (1994), 63--80.

Refereed conference proceedings

To appear
M. Eleuteri, Z. Farnana, O. E. Kansanen and R. Korte, Stability of solutions of the double obstacle problem on metric spaces, to appear in Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010).

2009
A. Björn and J. Björn, First-order Sobolev spaces on metric spaces, in Function Spaces, Inequalities and Interpolation (Paseky, 2009), pp. 1--29, Matfyzpress, Prague, 2009.
S. J. Gardiner and T. Sjödin, Partial balayage and the exterior inverse problem of potential theory, in Potential theory and stochastics in Albac, Theta Ser. Adv. Math. 11, pp. 111--123, Theta, Bucharest, 2009.
S. J. Gardiner and T. Sjödin, Potential theory in Denjoy domains, in Analysis and mathematical physics, pp. 143--166, Trends Math., Birkhäuser, Basel, 2009.

2006
J. Björn, Wiener criterion for Cheeger p-harmonic functions on metric spaces, in Potential Theory in Matsue, Advanced Studies in Pure Mathematics 44, pp. 103--115, Mathematical Society of Japan, Tokyo, 2006

2005
T. Sjödin, Quadrature identities and deformation of quadrature domains, in Quadrature Domains and their Applications, Operator Theory, Advances and Applications 156, pp. 239--255, Birkhäuser, Basel, 2005.

2003
A. Björn, p-harmonic measures and the Perron method for p-harmonic functions}, in Future Trends in Geometric Function Theory RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dep. Math. Stat. 92, pp. 23--29, Univ. Jyväskylä, Jyväskylä, 2003.
A. Björn, Removable singularities for analytic functions in Hardy spaces, BMO and locally Lipschitz spaces, in Progress in Analysis. Proceedings of the 3rd International ISAAC Congress (Berlin, 2001), vol. 1, pp. 445--450, World Scientific, Singapore, 2003.
J. Björn, Dirichlet problem for p-harmonic functions in metric spaces, in Future Trends in Geometric Function Theory RNC Workshop Jyväskylä 2003, Rep. Univ. Jyväskylä Dep. Math. Stat. 92, pp. 31--38, Univ. Jyväskylä, Jyväskylä, 2003.

1996
A. Björn, Removable singularities for Hardy spaces in subdomains of C, in Potential Theory - ICPT 94 (J. Král, J. Lukes, I. Netuka and J. and Veselý, eds.), pp. 287--295, de Gruyter, Berlin-New York, 1996.
A. Björn, Some open problems relating removable singularities for Hardy spaces and Hausdorff measures, in Potential Theory - ICPT 94 (J. Král, J. Lukes, I. Netuka and J. and Veselý, eds.), p. 474, de Gruyter, Berlin-New York, 1996

1993
H. Riesel and A. Björn, Generalized Fermat numbers, in Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics (W.Gautschi, ed.), Proc. Symp. Appl. Math. 48, pp. 583--587, Amer. Math. Soc., Providence, RI, 1994.

Popular science articles

A. Björn, Why is there no Nobel prize in mathematics?, De Morgan Association Newsletter, Issue 11, Dept. of Maths., University College London, London, 2003/04.

Master's thesis

Hannes Uppman, The Reflection Principle for One-dimensional Quasiminimizers, 2009 (supervisor A. Björn).
Tomas Andersson, An iterative solution method for p-harmonic functions on finite graphs with an implementation, 2009 (supervisor A. Björn).
John Karlsson, Lebesgue points, Hölder continuity and Sobolev functions, 2008 (supervisor J. Björn).
Patrik Leifson, Fractal sets and dimensions, 2006 (supervisor J. Björn).
David Färm, Upper gradients and Sobolev spaces on metric spaces, 2006 (supervisor J. Björn).
Lisa Hallingström, Primkontroll av tal på formen $k \cdot 2^q+1$ med program i Fortran 77, 2003 (supervisor A. Björn).
Svante Landgraf, Dominating sets for real parts of holomorphic functions, 2003 (supervisor A. Björn).

Bachelor's thesis

Jimmie Enhäll, Ett problem inom talteori, 2005 (supervisor A. Björn).
Tobias Svensson, Common factors in generalized Fermat numbers, 2005 (supervisor A. Björn).