Search results for "Mathematica"
showing 10 items of 7971 documents
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.
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…
Uniform, Sobolev extension and quasiconformal circle domains
1991
This paper contributes to the theory of uniform domains and Sobolev extension domains. We present new features of these domains and exhibit numerous relations among them. We examine two types of Sobolev extension domains, demonstrate their equivalence for bounded domains and generalize known sufficient geometric conditions for them. We observe that in the plane essentially all of these domains possess the trait that there is a quasiconformal self-homeomorphism of the extended plane which maps a given domain conformally onto a circle domain. We establish a geometric condition enjoyed by these plane domains which characterizes them among all quasicircle domains having no large and no small bo…
Embedding of Sobolev Spaces into Lipschitz Spaces
1989
The main result of the paper is that if Ω is a bounded uniform domain in ℝn and p>n, then the Sobolev space Wl, p(Ω) embeds continously into Cα(Ω), α = 1 - n/p.
An Itô Formula for rough partial differential equations and some applications
2020
AbstractWe investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form $\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u$ ∂ t u − A t u − f = ( X ̇ t ( x ) ⋅ ∇ + Y ̇ t ( x ) ) u on $[0,T]\times \mathbb {R}^{d}.$ [ 0 , T ] × ℝ d . To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove a…