We develop a framework for studying variational problems in Banach spaces with respect to gradien... more 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.
Annales De L Institut Henri Poincare Analyse Non Lineaire, Nov 1, 2010
In this paper we use quasiminimizing properties of radial power-type functions to deduce countere... more In this paper we use quasiminimizing properties of radial power-type functions to deduce counterexamples to certain Caccioppoli-type inequalities and weak Harnack inequalities for quasisuperharmonic functions, both of which are well-known to hold for p-superharmonic functions. We also obtain new bounds on the local integrability for quasisuperharmonic functions. Furthermore we show that the logarithm of a positive quasisuperminimizer has bounded mean oscillation and belongs to a Sobolev type space.
We study local connectedness, local accessibility and finite connectedness at the boundary, in re... more 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.
We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions... more We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(\cdot)$-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded $p(\cdot)$-harmonic functions and give some new characterizations of $W^{1, p(\cdot)}_0$ spaces. We also show that $p(\cdot)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.
We extend a result of John Lewis [L] by showing that ifa doubling metric measure space supports a... more We extend a result of John Lewis [L] by showing that ifa doubling metric measure space supports a (1, q0)-Poincare inequality for some 1 < qo < p, then every uniformly p-fat set is uniformly q-fat for some q < p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation of pharmonic functions and p-energy minimizers near a boundary point. Sweden. We wish to thank both institutes for their support. The first author was also supported by grants from the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation. We also thank Juha Kinnunen for pointing out the reference [Mi] and for other useful discussions. We also wish to thank Andreas Wannebo for interesting discussions related to his work and Juha Heinonen for his encouragement.
ABSTRACT We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and... more ABSTRACT We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R^n. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R^n, which are based on quasiconformality of radial stretchings in R^n.
We study the p-fine topology on complete metric spaces equipped with a doubling measure supportin... more We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. 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.
We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E... more 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 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. Most of the results are new for open E (apart from those which are trivial in this case) and also on R n .
We study when characteristic and Hölder continuous functions are traces of Sobolev functions on d... more We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.
ABSTRACT We show that on complete doubling metric measure spaces X supporting a Poincaré inequali... more ABSTRACT We show that on complete doubling metric measure spaces X supporting a Poincaré inequality, all Newton–Sobolev functions u are quasicontinuous, i.e., that for every ε&gt;0 there is an open set U⊂X such that C p (U)&lt;ε and the restriction of u to X∖Uis continuous. This implies that the capacity is an outer capacity.
We prove the Cartan and Choquet properties for the fine topology on a complete metric space equip... more We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.
In this paper we develop the Perron method for solving the Dirichlet problem for the analog of th... more 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.
Let ui be a Qi-quasisuperminimizer, i = 1, 2, and u = min{u1, u2}, where 1 ≤ Q1 ≤ Q2. Then u is a... more Let ui be a Qi-quasisuperminimizer, i = 1, 2, and u = min{u1, u2}, where 1 ≤ Q1 ≤ Q2. Then u is a quasisuperminimizer, and we improve upon the known upper bound (due to Kinnunen and Martio) for the optimal quasisuperminimizing constant Q of u. We give the first examples with Q > Q2, and show that in general Q > Q2 whenever Q1 > 1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.
Commentationes Mathematicae Universitatis Carolinae
Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincar... more Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincare inequality. We show that open subsets of X can be approx- imated by regular sets. This has applications in nonlinear potential theory on metric spaces. InparticularitmakesitpossibletodefineWienersolutionsoftheDirichletprob- lem for p-harmonic functions and to show that they coincide with three other notions of generalized solutions.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2014
Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this set... more Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.
ABSTRACT In this paper we develop the Perron method for solving the Dirichlet problem for the ana... more ABSTRACT 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.
We develop a framework for studying variational problems in Banach spaces with respect to gradien... more 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.
Annales De L Institut Henri Poincare Analyse Non Lineaire, Nov 1, 2010
In this paper we use quasiminimizing properties of radial power-type functions to deduce countere... more In this paper we use quasiminimizing properties of radial power-type functions to deduce counterexamples to certain Caccioppoli-type inequalities and weak Harnack inequalities for quasisuperharmonic functions, both of which are well-known to hold for p-superharmonic functions. We also obtain new bounds on the local integrability for quasisuperharmonic functions. Furthermore we show that the logarithm of a positive quasisuperminimizer has bounded mean oscillation and belongs to a Sobolev type space.
We study local connectedness, local accessibility and finite connectedness at the boundary, in re... more 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.
We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions... more We study various boundary and inner regularity questions for $p(\cdot)$-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for $p(\cdot)$-harmonic functions into three disjoint classes: regular, semiregular and strongly irregular points. Regular and especially semiregular points are characterized in many ways. The discussion is illustrated by examples. Along the way, we present a removability result for bounded $p(\cdot)$-harmonic functions and give some new characterizations of $W^{1, p(\cdot)}_0$ spaces. We also show that $p(\cdot)$-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.
We extend a result of John Lewis [L] by showing that ifa doubling metric measure space supports a... more We extend a result of John Lewis [L] by showing that ifa doubling metric measure space supports a (1, q0)-Poincare inequality for some 1 < qo < p, then every uniformly p-fat set is uniformly q-fat for some q < p. This bootstrap result implies the Hardy inequality for Newtonian functions with zero boundary values for domains whose complements are uniformly fat. While proving this result, we also characterize positive Radon measures in the dual of the Newtonian space using the Wolff potential and obtain an estimate for the oscillation of pharmonic functions and p-energy minimizers near a boundary point. Sweden. We wish to thank both institutes for their support. The first author was also supported by grants from the Swedish Natural Science Research Council and the Knut and Alice Wallenberg Foundation. We also thank Juha Kinnunen for pointing out the reference [Mi] and for other useful discussions. We also wish to thank Andreas Wannebo for interesting discussions related to his work and Juha Heinonen for his encouragement.
ABSTRACT We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and... more ABSTRACT We obtain estimates for the nonlinear variational capacity of annuli in weighted R^n and in metric spaces. We introduce four different (pointwise) exponent sets, show that they all play fundamental roles for capacity estimates, and also demonstrate that whether an end point of an exponent set is attained or not is important. As a consequence of our estimates we obtain, for instance, criteria for points to have zero (resp. positive) capacity. Our discussion holds in rather general metric spaces, including Carnot groups and many manifolds, but it is just as relevant on weighted R^n. Indeed, to illustrate the sharpness of our estimates, we give several examples of radially weighted R^n, which are based on quasiconformality of radial stretchings in R^n.
We study the p-fine topology on complete metric spaces equipped with a doubling measure supportin... more We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincare inequality, 1 < p< oo. 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.
We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E... more 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 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. Most of the results are new for open E (apart from those which are trivial in this case) and also on R n .
We study when characteristic and Hölder continuous functions are traces of Sobolev functions on d... more We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.
ABSTRACT We show that on complete doubling metric measure spaces X supporting a Poincaré inequali... more ABSTRACT We show that on complete doubling metric measure spaces X supporting a Poincaré inequality, all Newton–Sobolev functions u are quasicontinuous, i.e., that for every ε&gt;0 there is an open set U⊂X such that C p (U)&lt;ε and the restriction of u to X∖Uis continuous. This implies that the capacity is an outer capacity.
We prove the Cartan and Choquet properties for the fine topology on a complete metric space equip... more We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.
In this paper we develop the Perron method for solving the Dirichlet problem for the analog of th... more 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.
Let ui be a Qi-quasisuperminimizer, i = 1, 2, and u = min{u1, u2}, where 1 ≤ Q1 ≤ Q2. Then u is a... more Let ui be a Qi-quasisuperminimizer, i = 1, 2, and u = min{u1, u2}, where 1 ≤ Q1 ≤ Q2. Then u is a quasisuperminimizer, and we improve upon the known upper bound (due to Kinnunen and Martio) for the optimal quasisuperminimizing constant Q of u. We give the first examples with Q > Q2, and show that in general Q > Q2 whenever Q1 > 1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.
Commentationes Mathematicae Universitatis Carolinae
Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincar... more Let X be a complete metric space equipped with a doubling Borel measure supporting a weak Poincare inequality. We show that open subsets of X can be approx- imated by regular sets. This has applications in nonlinear potential theory on metric spaces. InparticularitmakesitpossibletodefineWienersolutionsoftheDirichletprob- lem for p-harmonic functions and to show that they coincide with three other notions of generalized solutions.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2014
Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this set... more Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit smoothness exponent. Further, we study quasisymmetries between the boundaries of two trees, and show that they have rough quasiisometric extensions to the trees. Conversely, we show that every rough quasiisometry between two trees extends as a quasisymmetry between their boundaries. In both directions we give sharp estimates for the involved constants. We use this to obtain quasisymmetric invariance of certain Besov spaces of functions on Cantor type sets.
ABSTRACT In this paper we develop the Perron method for solving the Dirichlet problem for the ana... more ABSTRACT 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.
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Papers by Jana Bjorn