We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincare ... more We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincare inequality. In particular, we are interested in continuity and p- harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.
We note that three factors are missing from in Factors of generalized Fermat numbers by A. Björn ... more We note that three factors are missing from in Factors of generalized Fermat numbers by A. Björn and H. Riesel published in Math. Comp. 67 (1998), 441-446. In in there are unfortunately three factors missing. Fougeron [3] discovered that 31246291 · 2 52 + 1 11 2 51 + 1 and 33797295 · 2 65 + 1 10 2 62 + 3 2 62 .
Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. The... more Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. 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>Q_2, and show that in general Q>Q_2 whenever Q_1 >1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2014
ABSTRACT Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In... more ABSTRACT 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.
In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topo... more In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topologist's comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.
Mathematical Proceedings of the Royal Irish Academy, 2002
A set E⫅⊆⊊⊇ Ω is holomorphically dominating for if sup zϵE| f (z)|= sup zϵE| f (z)| for all holom... more A set E⫅⊆⊊⊇ Ω is holomorphically dominating for if sup zϵE| f (z)|= sup zϵE| f (z)| for all holomorphic functions on Ω. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point*(of Ω) from Ω\ Ē. Moreover, it ...
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.
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 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 prove the nonlinear fundamental convergence theorem for superharmonic functions on metric meas... more We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have L q -Lebesgue points everywhere.
We pursue a systematic treatment of the variational capacity on metric spaces and give full proof... more We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R n it is shown to coincide with the usual variational capacity considered in the literature.
We characterize regular boundary points for p-harmonic functions using weak barriers. We use this... more We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for A-harmonic functions under the usual assumptions on A, and for Cheeger p-harmonic functions in metric spaces.
We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré ... more We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and pharmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.
In this paper we study removable singularities for holomorphic functions such that sup z∈Ω |f (n)... more In this paper we study removable singularities for holomorphic functions such that sup z∈Ω |f (n) (z)| dist(z, ∂Ω) s < ∞. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. , and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on Ω and be holomorphic on Ω E, whereas we only assume that the functions belong to the function space on Ω E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. obtained some results for s = 0.
We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincare ... more We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincare inequality. In particular, we are interested in continuity and p- harmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.
We note that three factors are missing from in Factors of generalized Fermat numbers by A. Björn ... more We note that three factors are missing from in Factors of generalized Fermat numbers by A. Björn and H. Riesel published in Math. Comp. 67 (1998), 441-446. In in there are unfortunately three factors missing. Fougeron [3] discovered that 31246291 · 2 52 + 1 11 2 51 + 1 and 33797295 · 2 65 + 1 10 2 62 + 3 2 62 .
Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. The... more Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. 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>Q_2, and show that in general Q>Q_2 whenever Q_1 >1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.
Journal für die reine und angewandte Mathematik (Crelles Journal), 2014
ABSTRACT Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In... more ABSTRACT 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.
In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topo... more In this paper we study the Perron method for solving the p-harmonic Dirichlet problem on the topologist's comb. For functions which are bounded and continuous at the accessible points, we obtain invariance of the Perron solutions under arbitrary perturbations on the set of inaccessible points. We also obtain some results allowing for jumps and perturbations at a countable set of points.
Mathematical Proceedings of the Royal Irish Academy, 2002
A set E⫅⊆⊊⊇ Ω is holomorphically dominating for if sup zϵE| f (z)|= sup zϵE| f (z)| for all holom... more A set E⫅⊆⊊⊇ Ω is holomorphically dominating for if sup zϵE| f (z)|= sup zϵE| f (z)| for all holomorphic functions on Ω. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point*(of Ω) from Ω\ Ē. Moreover, it ...
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.
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 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 prove the nonlinear fundamental convergence theorem for superharmonic functions on metric meas... more We prove the nonlinear fundamental convergence theorem for superharmonic functions on metric measure spaces. Our proof seems to be new even in the Euclidean setting. The proof uses direct methods in the calculus of variations and, in particular, avoids advanced tools from potential theory. We also provide a new proof for the fact that a Newtonian function has Lebesgue points outside a set of capacity zero, and give a sharp result on when superharmonic functions have L q -Lebesgue points everywhere.
We pursue a systematic treatment of the variational capacity on metric spaces and give full proof... more We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R n it is shown to coincide with the usual variational capacity considered in the literature.
We characterize regular boundary points for p-harmonic functions using weak barriers. We use this... more We characterize regular boundary points for p-harmonic functions using weak barriers. We use this to obtain some consequences on boundary regularity. The results also hold for A-harmonic functions under the usual assumptions on A, and for Cheeger p-harmonic functions in metric spaces.
We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré ... more We develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré inequality. In particular, we are interested in continuity and pharmonicity of the balayage. We also study connections to the obstacle problem. As applications, we characterize regular boundary points and polar sets in terms of balayage.
In this paper we study removable singularities for holomorphic functions such that sup z∈Ω |f (n)... more In this paper we study removable singularities for holomorphic functions such that sup z∈Ω |f (n) (z)| dist(z, ∂Ω) s < ∞. Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. , and Nguyen (1979) characterized removable singularities for some of these spaces. However, they used a different removability concept than in this paper. They assumed the functions to belong to the function space on Ω and be holomorphic on Ω E, whereas we only assume that the functions belong to the function space on Ω E, and are holomorphic there. Koskela (1993) obtained some results for our type of removability, in particular he showed the usefulness of the Minkowski dimension. obtained some results for s = 0.
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Papers by Anders Björn