Distributions and Operators

By Gerd Grubb. 2008

391 citations recorded by the source corpus.

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In Applied Mathematics

Citations in the corpus

Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators (2013, neutral): “r all t∈ R. As KM one can for example take the Poisson operator ϕ→ u solving the Dirichlet problem for (1− ∆)M, (1− ∆)Mu= 0 in Rn +,Mu= ϕon Rn−1 (an elementary treatment of the case M= 1 is found in [G09], Ch. 9). We shall here use the closely related choice, cf. (1.1) (e+ is sometimes left out): (1”
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, neutral): “Choose a partition of unity ( χ i ) Ni =1 for supp( u ) subordinate to W , with supp( χ i ) ⊂ B (cid:15) ( x i ) ( x i ) for 1 ≤ i ≤ N , this possible by [24, Theorem 2.17].”
Boundary Value Problems, Weyl Functions, and Differential Operators (2020, neutral): “Γ0f = ι−τ̃Df = ι−τ̃Dfη, f = fD + fη, with respect to the decomposition (4), since ker(τ̃D) = ker(τD) = dom(TD) and since Γ0 : ker(Tmax − ηI) → L(2)(∂Ω) is a bijection, one can introduce the γ-field γ(z), z ∈ ρ(TD), corresponding to { L(2)(∂Ω),Γ0,Γ1 } via”
Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators (2014, methodology): “ under 1◦. For the problem (2.15) with freplaced by 0 and greplaced by g2 we take G2 = g2. Then PaG2 is C∞ on a neighborhood of Ω (by the pseudolocal property of pseudodiﬀerential operators, cf. e.g. [G09], p. 177), so the reduced problem has a solution u2 ∈ Ea(Ω), and the given problem then has the”
Causal interpretation of stochastic differential equations (2013, methodology): “ implies that for all f ∈ C2 0(R p), we have Xd i=1 (ai −a˜i) ∂f ∂xi (x) + 1 2 Xp i=1 Xp j=1 (bij −˜bij) ∂2f ∂xi∂xj (A.21) (x) = 0. Fix i ≤ p. Again applying the approximation results of Chapter 2 of [16], there exists f ∈ C2 0(Rp) such that f(y) = yi in a neighborhood of x, implying ai−˜ai = 0. As ”
Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples (2012, background): “[Brown, Grubb and Wood, 2009; Brown et al., 2008; Grubb, 2009; Malamud, 2010; Posilicano and Raimondi, 2009] and is already essentially contained in the classical paper [Grubb, 1968].”
Some Liouville theorems for the fractional Laplacian (2014, background): “Therefore, we conclude that u is a polynomial (see for instance [G]).”
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data (2013, background): “The operator LA,q, equipped with the domain D(LA,q) := {u ∈ H (4)(Ω) : u|∂Ω = (∆u)|∂Ω = 0} is an unbounded closed operator on L(2)(Ω) with purely discrete spectrum, see [5].”