Regularity of quasi-minimizers on metric spaces

Using the theory of Sobolev spaces on a metric measure space we are able to apply calculus of variations and define p-harmonic functions as minimizers of the p-Dirichlet integral. More generally, we study regularity properties of quasi-minimizers of p-Dirichlet integrals in a metric measure space. Applying the De Giorgi method we show that quasi-minimizers, and in particular p-harmonic functions, satisfy Harnack's inequality, the strong maximum principle, and are locally H?lder continuous, if the space is doubling and supports a Poincaré inequality. Received: 12 May 2000 / Revised version: 20 April 2001     

Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

A regularity classification of boundary points for p-harmonic functions and quasiminimizers

In this paper it is shown that irregular boundary points for p-harmonic functions as well as for quasiminimizers can be divided into semiregular and strongly irregular points with vastly different boundary behaviour. This division is emphasized by a large number of characterizations of semiregular points. The results hold in complete metric spaces equipped with a doubling measure supporting a Poincaré inequality. They also apply to Cheeger p-harmonic functions and in the Euclidean setting to A-harmonic functions, with the usual assumptions on A.     

Lipschitz-type functions on metric spaces

In order to find metric spaces X for which the algebra Lip^∗(X) of bounded Lipschitz functions on X determines the Lipschitz structure of X, we introduce the class of small-determined spaces. We show that this class includes precompact and quasi-convex metric spaces. We obtain several metric characterizations of this property, as well as some other characterizations given in terms of the uniform approximation and the extension of uniformly continuous functions. In particular we show that X is small-determined if and only if every uniformly continuous real function on X can be uniformly approximated by Lipschitz functions.     

The Perron method for p-harmonic functions 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 1?q<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.     

The φ-harmonic approximation and the regularity of φ-harmonic maps

We extend the p-harmonic approximation lemma proved by Duzaar and Mingione for p-harmonic functions to φ-harmonic functions, where φ is a convex function. The proof is direct and is based on the Lipschitz truncation technique. We apply the approximation lemma to prove Hölder continuity for the gradient of a solution of a φ-harmonic system with critical growth.     

Bessel Potentials in Ahlfors Regular Metric Spaces

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L^p. Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.     

Duality for Lipschitz p-summing operators

Building upon the ideas of R. Arens and J. Eells (1956) [1] we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet (1969) [3] and Saphar (1970) [14], whose duals are spaces of Lipschitz p-summing operators. In particular, we identify the dual of the space of Lipschitz p-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson (2009) [6] — and use it to give a new characterization of the non-linear concept of Lipschitz p-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté (1976) [11], whose duals are analogues of A. Pietsch?s (p,r,s)-summing operators (A. Pietsch, 1980 [12]). As a special case, we get a Lipschitz version of (q,p)-dominated operators.     

Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces

We study the asymptotic Dirichlet problem for p-harmonic functions in a very general setting of Gromov hyperbolic metric measure spaces.     

Regularity of Lipschitz free boundaries in two-phase problems for the p-Laplace operator

In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are C1,γ-smooth for some γ∈(0,1). As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain.     

p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

For p-harmonic functions on unweighted R², with 1<p<∞, we show that if the boundary values f has a jump at an (asymptotic) corner point z₀, then the Perron solution Pf is asymptotically a+barg(z−z₀)+o(|z−z₀|). We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p-harmonic measure of . We also obtain various invariance results for functions with jumps and perturbations on small sets. For p>2 these results are new also for continuous functions. Finally we look at generalizations to Rn and metric spaces.     

New explicit solutions to the p-Laplace equation based on isoparametric foliations

In contrast to an infinite family of explicit examples of two-dimensional p-harmonic functions obtained by G. Aronsson in the late 80s, there is very little known about the higher-dimensional case. In this paper, we show how to use isoparametric polynomials to produce diverse examples of p-harmonic and biharmonic functions. Remarkably, for some distinguished values of p and the ambient dimension n this yields first examples of rational and algebraic p-harmonic functions. Moreover, we show that there are no p-harmonic polynomials of the isoparametric type. This supports a negative answer to a question proposed in 1980 by J. Lewis.     

Quasiopen and <Emphasis Type="Italic">p</Emphasis>-Path Open Sets,and Characterizations of Quasicontinuity

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-Poincaré 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.     

Differentiability of cone-monotone functions on separable Banach space

Motivated by applications to (directionally) Lipschitz functions, we provide a general result on the almost everywhere Gâteaux differentiability of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone with non-empty interior. This seemingly arduous restriction is useful, since it covers the case of directionally Lipschitz functions, and necessary. We show by way of example that most results fail more generally.

     

On Arc-Wise Essentially Smooth Mappings Between Banach Spaces

Recently, Borwein and Moors introduced a new class of real-valued locally Lipschitz functions, that are similar in nature and definition to Valadier's saine functions, which they called arc-wise essentially smooth. They showed that if g n M is an arc-wise essentially smooth real-valued function and f m M n is strictly differentiable almost everywhere, then g f m M is also strictly differentiable almost everywhere. They also showed that this class possesses strong closure properties. In this paper, we give an appropriate extension of this class to locally Lipschitz mappings defined between Banach spaces. We show that the results established by Borwein and Moors in the finite dimensional setting also hold for arc-wise essentially smooth mappings defined between Banach spaces.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Integrability of rotationally symmetric n-harmonic maps

A rotationally symmetric n-harmonic map is a rotationally symmetric p-harmonic map between two n-dimensional model spaces such that p=n. We show that rotationally symmetric n-harmonic maps can be integrated and are n-harmonic diffeomorphism, and apply such results to investigate the asymptotic behaviors of these maps. We also derive this integrability using Lie theory.     

Necessity of a Wiener type condition for boundary regularity of quasiminimizers and nonlinear elliptic equations

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-Poincaré 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 .     

Linear relations for p-harmonic functions

We discuss the p-harmonicity of the linear combination of p-harmonic functions in the Euclidean space and on a tree. If p≠2, the p-harmonicity is non-linear, i.e., the linear combination of p-harmonic functions need not be p-harmonic. In spite of this non-linear nature, we find some p-harmonic functions whose linear combinations become p-harmonic.