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Anders Björn

Publications and source records attributed to Anders Björn.

At least 37 records · Page 2Linked to original sources
arXiv2021

Removable singularities for bounded $\mathcal{A}$-(super)harmonic and quasi(super)harmonic functions on weighted $\mathbf{R}^n$

It is well known that sets of $p$-capacity zero are removable for bounded $p$-harmonic functions, but on metric spaces there are examples of removable sets of positive capacity. In this paper, we show that this can happen even on unweighted $\mathbf{R}^n$ when $n > p$, although only in very special cases. A complete characterization of removable singularities for bounded $\mathcal{A}$-harmonic functions on weighted $\mathbf{R}^n$, $n \ge 1$, is also given, where the weight is $p$-admissible. The same characterization is also shown to hold for bounded quasiharmonic functions on weighted $\mathbf{R}^n$, $n \ge 2$, as well as on unweighted $\mathbf{R}$. For bounded $\mathcal{A}$-superharmonic functions and bounded quasisuperharmonic functions on weighted $\mathbf{R}^n$, $n \ge 2$, we show that relatively closed sets are removable if and only if they have zero capacity.

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arXiv2021

Volume growth, capacity estimates, $p$-parabolicity and sharp integrability properties of $p$-harmonic Green functions

In a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general $p$-harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted $\mathbf{R}^n$ and on manifolds. The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for $p$-harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the $p$-parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

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arXiv2020

Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Let $X$ be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion $\widehat{X}$ of $X$ and use them to obtain several results on $X$ itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to $p$-harmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.

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arXiv2020

Resolutivity and invariance for the Perron method for degenerate equations of divergence type

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u) = 0$ in a bounded open set $Ω\subset\mathbf{R}^n$. The vector-valued function $\mathcal{A}$ satisfies the standard ellipticity assumptions with a parameter $1<p<\infty$ and a $p$-admissible weight $w$. We show that arbitrary perturbations on sets of $(p,w)$-capacity zero of continuous (and certain quasicontinuous) boundary data $f$ are resolutive and that the Perron solutions for $f$ and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of $(p,w)$-capacity zero.

math.AP↗
arXiv2019

Semiregular and strongly irregular boundary points for $p$-harmonic functions on unbounded sets in metric spaces

The trichotomy between regular, semiregular, and strongly irregular boundary points for $p$-harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $p$-harmonic measures, removability, and semibarriers.

math.AP↗
arXiv2019

Existence and almost uniqueness for $p$-harmonic Green functions on bounded domains in metric spaces

We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted $\mathbf{R}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, or under similar local assumptions.

math.AP↗
arXiv2019

Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

math.AP↗
arXiv2019

The quasisuperminimizing constant for the minimum of two quasisuperminimizers in R^n

It was shown in Björn--Björn--Korte ("Minima of quasisuperminimizers", Nonlinear Anal. 155 (2017), 264-284) that $u:=\min\{u_1,u_2\}$ is a $\overline{Q}$-quasisuperminimizer if $u_1$ and $u_2$ are $Q$-quasisuperminimizers and $\overline{Q}=2Q^2/(Q+1)$. Moreover, one-dimensional examples therein show that $\overline{Q}$ is close to optimal. In this paper we give similar examples in higher dimensions. The case when $u_1$ and $u_2$ have different quasisuperminimizing constants is considered as well.

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arXiv2018

The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications

In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.

math.AP↗
arXiv2018

The Liouville theorem for $p$-harmonic functions and quasiminimizers with finite energy

We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite $p$th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $p$-Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $\mathbf{R}$, we characterize all locally doubling measures, supporting a local $p$-Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $p$-energy, and show that a quasiminimizer is of finite $p$-energy if and only if it is bounded. As $p$-harmonic functions are quasiminimizers they are covered by these results.

math.MG↗
arXiv2018

Local and semilocal Poincaré inequalities on metric spaces

We consider several local versions of the doubling condition and Poincaré inequalities on metric spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincaré inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward. We also conclude that many qualitative, as well as quantitative, properties of p-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions.

math.AP↗
arXiv2018

A uniqueness result for functions with zero fine gradient on quasiconnected and finely connected sets

We show that every Sobolev function in $W^{1,p}_{\textrm{loc}}(U)$ on a $p$-quasiopen set $U \subset {\bf R}^n$ with a.e.-vanishing $p$-fine gradient is a.e.-constant if and only if $U$ is $p$-quasiconnected. To prove this we use the theory of Newtonian Sobolev spaces on metric measure spaces, and obtain the corresponding equivalence also for complete metric spaces equipped with a doubling measure supporting a $p$-Poincaré inequality. On unweighted ${\bf R}^n$, we also obtain the corresponding result for $p$-finely open sets in terms of $p$-fine connectedness, using a deep result by Latvala.

math.AP↗
arXiv2018

Boundary regularity for the porous medium equation

We study the boundary regularity of solutions to the porous medium equation $u_t = Δu^m$ in the degenerate range $m>1$. In particular, we show that in cylinders the Dirichlet problem with positive continuous boundary data on the parabolic boundary has a solution which attains the boundary values, provided that the spatial domain satisfies the elliptic Wiener criterion. This condition is known to be optimal, and it is a consequence of our main theorem which establishes a barrier characterization of regular boundary points for general -- not necessarily cylindrical -- domains in ${\bf R}^{n+1}$. One of our fundamental tools is a new strict comparison principle between sub- and superparabolic functions, which makes it essential for us to study both nonstrict and strict Perron solutions to be able to develop a fruitful boundary regularity theory. Several other comparison principles and pasting lemmas are also obtained. In the process we obtain a rather complete picture of the relation between sub/super\-para\-bolic functions and weak sub/super\-solu\-tions.

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arXiv2017

The tusk condition and Petrovski criterion for the normalized $p\mspace{1mu}$-parabolic equation

We study boundary regularity for the normalized $p\mspace{1mu}$-parabolic equation in arbitrary bounded domains. Effros and Kazdan (Indiana Univ. Math. J. 20 (1970), 683-693) showed that the so-called tusk condition guarantees regularity for the heat equation. We generalize this result to the normalized $p\mspace{1mu}$-parabolic equation, and also obtain Hölder continuity. The tusk condition is a parabolic version of the exterior cone condition. We also obtain a sharp Petrovski criterion for the regularity of the latest moment of a domain. This criterion implies that the regularity of a boundary point is affected if one side of the equation is multiplied by a constant.

math.AP↗
arXiv2017

Tensor products and sums of p-harmonic functions, quasiminimizers and p-admissible weights

The tensor product of two p-harmonic functions is in general not p-harmonic, but we show that it is a quasiminimizer. More generally, we show that the tensor product of two quasiminimizers is a quasiminimizer. Similar results are also obtained for quasisuperminimizers and for tensor sums. This is done in weighted R^n with p-admissible weights. It is also shown that the tensor product of two p-admissible measures is p-admissible. This last result is generalized to metric spaces.

math.AP↗
arXiv2016

Dichotomy of global capacity density in metric measure spaces

The variational capacity cap_p in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every subset E of R^n, inf_{x in R^n} (cap_p(E \cap B(x,r),B(x,2r)) / cap_p(B(x,r),B(x,2r))) is either zero or tends to 1 as r tends to infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R^n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

math.AP↗
arXiv2016

The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

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