Traces of functions of generalized Sobolev classes

Considering the Sobolev type function classes on a metric space equipped with a Borel measure we address the question of compactness of embeddings of the space of traces into Lebesgue spaces on the sets of less “dimension.” Also, we obtain compactness conditions for embeddings of the traces of the classical Sobolev spaces W _p ¹ on the “zero” cusp with a Hölder singularity at the vertex.     

Traces of Sobolev functions with one square integrable directional derivative

We consider the Sobolev spaces of square integrable functions v, from ?ⁿ or from one of its hyperquadrants Q, into a complex separable Hilbert space, with square integrable sum of derivatives ∑?_??v. In these spaces we define closed trace operators on the boundaries ?Q and on the hyperplanes {r_?? = z}, z ∈ ?\{0}, which turn out to be possibly unbounded with respect to the usual L²‐norm for the image. Therefore, we also introduce bigger trace spaces with weaker norms which allow to get bounded trace operators, and, even if these traces are not L², we prove an integration by parts formula on each hyperquadrant Q. Then we discuss surjectivity of our trace operators and we establish the relation between the regularity properties of a function on ?ⁿ and the regularity properties of its restrictions to the hyperquadrants Q. Copyright © 2005 John Wiley & Sons, Ltd.     

Traces of Sobolev functions on the Ahlfors sets of Carnot groups

We prove the converse of the trace theorem for the functions of the Sobolev spaces W _p ^l on a Carnot group on the regular closed subsets called Ahlfors d-sets (the direct trace theorem was obtained in one of our previous publications). The theorem generalizes Johnsson and Wallin’s results for Sobolev functions on the Euclidean space. As a consequence we give a theorem on the boundary values of Sobolev functions on a domain with smooth boundary in a two-step Carnot group. We consider an example of application of the theorems to solvability of the boundary value problem for one partial differential equation.     

Traces and Sobolev extension domains

Assume that is a bounded domain and its boundary is -regular, . We show that if there exists a bounded trace operator , and , and -Hölder continuous functions are dense in , , then the domain is a -extension domain.

     

On the boundary limits of monotone Sobolev functions in variable exponent Orlicz spaces

Our aim in this note is to deal with boundary limits of monotone Sobolev functions with ▽u∈ Lp(·)logLq(·)(B) for the unit ball BRn. Here p(·) and q(·) are variable exponents satisfyingthe log-Hlder and the log log-Hlder conditions, respectively.     

Boundary behavior of monotone Sobolev functions on John domains in a metric space

This article deals with weighted boundary limits of monotone Sobolev functions on bounded s-John domains in a metric space.     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

Traces of oscillating functions

We show that the trace of an indefinitely oscillating function on a subspace of ^d is not always indefinitely oscillating. In the periodic case, the number of oscillations of the trace depends on the regularity of the function. In the general case, we exhibit a definitive counter-example.     

Traces of functions belonging to Sobolev and Besov spaces and extensions from subsets of the Euclidean space

It is proved that from the existence of the trace operator, 0n and E is a Borel subset of ⁿ.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, AN SSSR, Vol. 157, pp. 137–145, 1987.     

Weakly monotone functions

The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spacesW ^1,p ,p >n ? 1. It is proven that such weakly monotone functions are continuous except in a singular set ofp-capacity zero that is empty in the casep =n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.     

Generalized absolutely monotone functions

The concept of absolutely monotone functions is generalized by replacing the conditionsφ ^(k)(t)≧0,k=0, 1, … by an infinite sequence of differential inequalitiesφ(t)≧0,L _kφ(t)≧0,k=1, 2, …, where theL _k are differential operators of a special type. It is shown that these functions have a valid series expansion in terms of basic functions associated with the operatorsL _k.     

Almost Sharp Descriptions of Traces of Sobolev Spaces on Compacta

Mathematical Notes -     

Borderline Case of Traces and Extensions for Weighted Sobolev Spaces

In this paper,we study the traces and the extensions for weighted Sobolev spaces on upper half spaces when the weights reach to the borderline cases.We first give a full characterization of the existence of trace spaces for these weighted Sobolev spaces,and then study the trace parts and the extension parts between the weighted Sobolev spaces and a new kind of Besov-type spaces(on hyperplanes) which are defined by using integral averages over selected layers of dyadic cubes.