Search results for "Sobolev Space"
showing 10 items of 164 documents
Regularity of the inverse of a Sobolev homeomorphism in space
2006
Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.
Removable sets for Sobolev spaces
1999
We study removable sets for the Sobolev space W1,p. We show that removability for sets lying in a hyperplane is essentially determined by their thickness measured in terms of a concept of p-porosity.
Invertibility of Sobolev mappings under minimal hypotheses
2010
Abstract We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W 1 , n mapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion. The integrability assumption is shown to be optimal.
Maximal potentials, maximal singular integrals, and the spherical maximal function
2014
We introduce a notion of maximal potentials and we prove that they form bounded operators from L to the homogeneous Sobolev space Ẇ 1,p for all n/(n − 1) < p < n. We apply this result to the problem of boundedness of the spherical maximal operator in Sobolev spaces.
Regularity of the Inverse of a Sobolev Homeomorphism
2011
We give necessary and sufficient conditions for the inverse ofa Sobolev homeomorphism to be a Sobolev homeomorphism and conditions under which the inverse is of bounded variation.
Generalized dimension estimates for images of porous sets under monotone Sobolev mappings
2014
We give an essentially sharp estimate in terms of generalized Hausdorff measures for images of porous sets under monotone Sobolev mappings, satisfying suitable Orlicz-Sobolev conditions.
Continuity of the maximal operator in Sobolev spaces
2006
We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces W 1,p (R n ), 1 < p < ∞. As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.
REGULARITY OF THE FRACTIONAL MAXIMAL FUNCTION
2003
The purpose of this work is to show that the fractional maximal operator has somewhat unexpected regularity properties. The main result shows that the fractional maximal operator maps -spaces boundedly into certain first-order Sobolev spaces. It is also proved that the fractional maximal operator preserves first-order Sobolev spaces. This extends known results for the Hardy–Littlewood maximal operator.
First-Order Calculus on Metric Measure Spaces
2020
In this chapter we develop a first-order differential structure on general metric measure spaces. First of all, the key notion of cotangent module is obtained by combining the Sobolev calculus (discussed in Chap. 2) with the theory of normed modules (described in Chap. 3). The elements of the cotangent module L2(T∗X), which are defined and studied in Sect. 4.1, provide a convenient abstraction of the concept of ‘1-form on a Riemannian manifold’.
Penalty Function Methods for the Numerical Solution of Nonlinear Obstacle Problems with Finite Elements
2008
A class of penalty function methods for the solution of nonlinear variational inequalities with obstacles ⩽ 0 fur alle v ⩾ ψ in the Sobolev space W1, p (ω) is studied. The (nonlinear) penalty equations are solved by finite element techniques; the order of convergence of this procedure which depends on the regularity of the solution as well as on the finite elements used is investigated. Eine Klasse von Penalty-Methoden zur Losung nichtlinearer Variationsungleichungen mit Hindernisnebenbedingungen ⩽ 0 fur alle v ⩾ ψ im Sobolev Raum W1, p (ω) wird untersucht. Die (nichtlinearen) Penalty-Gleichungen werden mit Hilfe der Finite Elemente Methode gelost; die Konvergenzordnung dieses Verfahrens, w…