We study various boundary and inner regularity questions for p(.)-(super)harmonic functions in Euclidean domains. In particular, we prove the Kellogg property and introduce a classification of boundary points for p(.)-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(.)-harmonic functions and give some new characterizations of W-0(1,p(.)) spaces. We also show that p(.)-superharmonic functions are lower semicontinuously regularized, and characterize them in terms of lower semicontinuously regularized supersolutions.
In this paper we propose a new definition of prime ends for domains in metric spaces under rather general assumptions. We compare our prime ends to those of Caratheodory and Nakki. Modulus ends and prime ends, defined by means of the p-modulus of curve families, are also discussed and related to the prime ends. We provide characterizations of singleton prime ends and relate them to the notion of accessibility of boundary points, and introduce a topology on the prime end boundary. We also study relations between the prime end boundary and the Mazurkiewicz boundary. Generalizing the notion of John domains, we introduce almost John domains, and we investigate prime ends in the settings of John domains, almost John domains and domains which are finitely connected at the boundary.
The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare 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.
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.
We use variational methods to obtain a pointwise estimate near a boundary point for quasisubminimizers of the p-energy integral and other integral functionals in doubling metric measure spaces admitting a p-Poincar, inequality. It implies a Wiener type condition necessary for boundary regularity for p-harmonic functions on metric spaces, as well as for (quasi)minimizers of various integral functionals and solutions of nonlinear elliptic equations on R (n) .
We study p-harmonic functions in complete metric spaces equipped with a doubling Borel measure supporting a weak (1, p)-Poincaré inequality, 1 < p < ∞. We establish the barrier classification of regular boundary points from which it also follows that regularity is a local property of the boundary. We also prove boundary regularity at the fixed (given) boundary for solutions of the one-sided obstacle problem on bounded open sets. Regularity is further characterized in several other ways. Our results apply also to Cheeger p-harmonic functions and in the Euclidean setting to script A sign-harmonic functions, with the usual assumptions on script A sign.
We consider several local versions of the doubling condition and Poincare inequalities on metric measure 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 Poincare 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. (C) 2018 Elsevier Masson SAS. All rights reserved.
The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing. Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups. Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theories.
This monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers. The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher. The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysis.
The first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces. In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric space.
Each chapter contains historical notes with relevant references and an extensive index is provided at the end of the book.
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.
Let X be a noncomplete metric measure space satisfying the usual (local) assumptions of a doubling property and a Poincare inequality. We study extensions of Newtonian Sobolev functions to the completion (X) over cap 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 Poincare 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. (C) 2018 Elsevier Inc. All rights reserved.
In this paper we examine the quasiminimizing properties of radial power-type functions u(x) = vertical bar x vertical bar(alpha) in R-n. We find the optimal quasiminimizing constant whenever u is a quasiminfinizer of the p-Dirichlet integral, p not equal n, and similar results when u is a quasisub- and quasisuperminimizer. We also obtain similar results for log-powers when p = n.
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.
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. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.
In this paper we obtain sharp Petrovskii criteria for the p-parabolic equation, both in the degenerate case p amp;gt; 2 and the singular case 1 amp;lt; p amp;lt; 2 We also give an example of an irregular boundary point at which there is a barrier, thus showing that regularity cannot be characterized by the existence of just one barrier.
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.
We study the boundary regularity of solutions to the porous medium equation in the degenerate range . 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 . 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/superparabolic functions and weak sub/supersolutions.
dUsing 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.
We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q-quasisuperminimizers. More precisely, if u(i) is a Q(i)-quasisuperminimizer, i = 1,2, where 1 amp;lt; Q(1) amp;lt; Q(2), then u = min{u(1), u(2)} is a Q-quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q. We provide the first examples of this phenomenon, i.e. that Q amp;gt; Q(2). In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q(1)Q(2) turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers. (C) 2017 Elsevier Ltd. All rights reserved.
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.
We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a p-Poincar, inequality, 1 amp;lt; p amp;lt; a. 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.
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.
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 obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar, inequality. In particular, if the measure has the 1-annular decay property at and the metric space supports a pointwise 1-Poincar, inequality at , then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at . This generalizes the known estimate for the usual variational capacity in unweighted . We also characterize the 1-annular decay property and provide examples which illustrate the sharpness of our results.
In this paper we give various characterizations of quasiopen sets and quasicontinuous functions on metric spaces. For complete metric spaces equipped with a doubling measure supporting a p-Poincar, inequality we show that quasiopen and p-path open sets coincide. Under the same assumptions we show that all Newton-Sobolev functions on quasiopen sets are quasicontinuous.
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 develop a theory of balayage on complete doubling metric measure spaces supporting a Poincaré 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 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.
A. Baernstein II (Comparison of p-harmonic measures of subsets of the unit circle, St. Petersburg Math. J. 9 (1998), 543-551, p. 548), posed the following question: If G is a union of m open arcs on the boundary of the unit disc D, then is w _{a,p}(G)=w _{a,p}(G), where w _{a,p} denotes the p-harmonic measure? (Strictly speaking he stated this question for the case m=2.) For p=2 the positive answer to this question is well known. Recall that for p≠2 the p-harmonic measure, being a nonlinear analogue of the harmonic measure, is not a measure in the usual sense.
The purpose of this note is to answer a more general version of Baernstein's question in the affirmative when 1G is the restriction to ∂D of a Sobolev function from W _{1,p}(C).
For p≥2 it is no longer true that X_{G} belongs to the trace class. Nevertheless, we are able to show equality for the case m=1 of one arc for all 1, using a very elementary argument. A similar argument is used to obtain a result for starshaped domains.
Finally we show that in a certain sense the equality holds for almost all relatively open sets.
We show that on complete doubling metric measure spaces X supporting a Poincare inequality, all Newton-Sobolev functions u are quasicontinuous, i.e. that for every epsilon > 0 there is an open set U subset of X such that C-p(U) < epsilon and the restriction of u to X\U is continuous. This implies that the capacity is an outer capacity.
We study when characteristic and Holder 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. ©Canadian Mathematical Society 2007.
We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a PoincarΘ inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity. We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p = 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p = 2, enabling us to extend the solvability of the problem to L1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p = 2. Finally, we give an estimate for the Hausdorff dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces.
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.
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 use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.
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.
One of the main break-throughs in the 20th century mathematics was De Giorgi's proof of Hölder continuity for solutions of elliptic PDEs. His method has since then been used to prove interior regularity in various contexts.It is maybe less known, though not entirely surprising that De Giorgi's method also yields sufficient conditions for boundary regularity. In the talk, I will discuss a recent variation of De Giorgi's method which goes in the opposite direction, leading to a necessary condition for boundary regularity of PDEs and variational integrals.