A discontinuous Sobolev function exists

2018 ◽  
Vol 147 (2) ◽  
pp. 637-639
Author(s):  
Przemysław Górka ◽  
Artur Słabuszewski
Keyword(s):  
Sobolev Function

scholarly journals Strongly nonlinear potential theory on metric spaces

2002 ◽  
Vol 7 (7) ◽  
pp. 357-374 ◽  
Author(s):  
Noureddine Aïssaoui
Keyword(s):  
Metric Spaces ◽  
Outer Measure ◽  
Hausdorff Measures ◽  
Sobolev Function ◽  
Nonlinear Potential Theory ◽  
Strongly Nonlinear ◽  
Capacity Theory ◽  
Convergence Results ◽  
Nonlinear Potential ◽  
Additional Regularity

We define Orlicz-Sobolev spaces on an arbitrary metric space with a Borel regular outer measure, and we develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. We prove that each Orlicz-Sobolev function has a quasi-continuous representative. We give estimates for the capacity of balls when the measure is doubling. Under additional regularity assumption on the measure, we establish some relations between capacity and Hausdorff measures.

scholarly journals Sobolev spaces with respect to a weighted Gaussian measure in infinite dimensions

2019 ◽  
Vol 22 (04) ◽  
pp. 1950026 ◽  
Author(s):  
S. Ferrari
Keyword(s):  
Banach Space ◽  
Sobolev Spaces ◽  
Gaussian Measure ◽  
Separable Banach Space ◽  
Positive Function ◽  
Infinite Dimensions ◽  
Sobolev Function ◽  
Gradient Operator ◽  
Divergence Operator ◽  
Sobolev Functions

Let [Formula: see text] be a separable Banach space endowed with a nondegenerate centered Gaussian measure [Formula: see text] and let [Formula: see text] be a positive function on [Formula: see text] such that [Formula: see text] and [Formula: see text] for some [Formula: see text] and [Formula: see text]. In this paper, we introduce and study Sobolev spaces with respect to the weighted Gaussian measure [Formula: see text]. We obtain results regarding the divergence operator (i.e. the adjoint in [Formula: see text] of the gradient operator along the Cameron–Martin space) and the trace of Sobolev functions on hypersurfaces [Formula: see text], where [Formula: see text] is a suitable version of a Sobolev function.

A strictly convex Sobolev function with null Hessian minors

2016 ◽  
Vol 55 (3) ◽  
Author(s):  
Zhuomin Liu ◽  
Jan Malý
Keyword(s):  
Sobolev Function ◽  
Strictly Convex

Fitting a Sobolev function to data I

2016 ◽  
Vol 32 (1) ◽  
pp. 275-376 ◽  
Author(s):  
Charles Fefferman ◽  
Arie Israel ◽  
Garving Luli
Keyword(s):  
Sobolev Function

Fitting a Sobolev function to data III

2016 ◽  
Vol 32 (3) ◽  
pp. 1039-1126 ◽  
Author(s):  
Charles Fefferman ◽  
Arie Israel ◽  
Garving Luli
Keyword(s):  
Sobolev Function

scholarly journals Weighted Variable Exponent Sobolev spaces on metric measure spaces

2018 ◽  
Vol 4 (2) ◽  
pp. 62-76
Author(s):  
Moulay Cherif Hassib ◽  
Youssef Akdim
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Metric Spaces ◽  
Maximal Inequality ◽  
Outer Measure ◽  
Sobolev Function ◽  
Convergence Results ◽  
Variable Exponent Sobolev Spaces ◽  
Measure Spaces ◽  
Weighted Variable

AbstractIn this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

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