Sobolev Spaces

By G. Burton. 2013

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In Pure Mathematics

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “erates spaces of differentiable functions known as the Sobolev spaces (Adams and Fournier 2003).”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, background): “It is also known that Bs2,2(Rd) = ( L2(Rd),W m2 (Rd) ) s/m,2 for any integer m > s (Adams and Fournier, 2003, Page 230).”
Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “To this end, we focus on the setting where X = R and H is a Sobolev space [1].”
On a new class of fractional partial differential equations (2014, background): “7 records the equivalence of X(R ) and L(R ), the following theorem from the book of Adams [1] allows us to understand the properties of this space.”
Landau Damping: Paraproducts and Gevrey Regularity (2013, methodology): “Proof It follows by induction on co-dimension with the standard H1/2 restriction theorem [1].”
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm (2019, background): “In particular, if c = k· ·kp for p 2 [1,1), then OT0 is the well-known p-Wasserstein distance [52].”
Sobolev Spaces on Non-Lipschitz Subsets of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^n$$\end{doc (2016, unknown): “(i) The equality (cid:101) H 0 (Ω) = ◦ H 0 (Ω) holds because the restriction operator is a unitary isomorphism from [2, Theorem 2.19].”
Porous media equations with two weights: smoothing and decay properties of energy solutions via Poincar\'e inequalities (2012, background): “∀r ∈ [1, 2) , ∀ε > 0 , ∀v ∈ L(Ω; ν) ∩W 1,2 0 (Ω; ν, μ) was already established in [29, Th.”
A projection-based stabilized finite element method for steady-state natural convection problem (2011, unknown): “The standard notations in [1] for Sobolev and Lebesgue spaces are used.”
Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity (2013, background): “Assumption 2 Ω ∪ΩI is bounded and satisfies the interior cone condition with parameters r0 > 0 and θ0 > 0, as defined by the property that (see [1] for details) for any point x ∈ Ω ∪ΩI , the intersection between the ball centered at x with radius r0 and the domain Ω ∪ΩI contains a cone with an angle”
Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori (2015, background): “Our references for the classical theory are [1, 42, 49, 53, 72, 73].”
Ricci curvature and $L^p$-convergence (2012, result): “Note that in [37] Kuwae-MachigashiraShioya showed a similar result about an explicit formula for the Dirichlet Laplacian on an Alexandrov space.”
Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data (2011, background): “The following theorem gives some properties of traces of functions living in W s, (Ω) ([1,22]).”
A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations (2011, unknown): “Throughout this section, denote by W k,l(Ω) the Sobolev space (see [1]), where l > 1 and k is a nonnegative integer.”
Functional analysis and exterior calculus on mixed-dimensional geometries (2017, unknown): “We refer the reader to more comprehensive texts for a complete introduction (some instructive and relevant works are [8,3,15,19,29,1], noting that in general throughout the text we will mostly be consistent with the notation as used in [3] with the exception that for a manifold Ωi, we denote the spa”