On regularized distance and related functions

1979 ◽  
Vol 83 (1-2) ◽  
pp. 115-122 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Continuous Function ◽  
Closed Subset ◽  
Parameter Family ◽  
Lipschitz Continuous Function ◽  
Partial Derivatives ◽  
Lipschitz Continuous ◽  
Order Of Magnitude ◽  
Derivatives Of ◽  
Smooth Approximations

SynopsisLetFbe any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complementcF, that approximates the distance fromFof any pointx ∈cFin the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

Approximation of Lipschitz Continuous Function on the Orlicz-Sobolev Space

2013 ◽  
Vol 15 (3) ◽  
pp. 251
Author(s):  
Chunfang LIU ◽  
Yongqiang FU ◽  
Yuesheng LUO ◽  
Shilei ZHANG
Keyword(s):  
Continuous Function ◽  
Sobolev Space ◽  
Lipschitz Continuous Function ◽  
Lipschitz Continuous

RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS

2005 ◽  
Vol 03 (02) ◽  
pp. 99-117 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
CRISTINEL MARDARE
Keyword(s):  
Positive Definite ◽  
Symmetric Matrices ◽  
Partial Derivatives ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
The Matrix ◽  
Connected Open Subset ◽  
Matrix Fields ◽  
Simply Connected ◽  
Derivatives Of

If a field A of class [Formula: see text] of positive-definite symmetric matrices of order two and a field B of class [Formula: see text] of symmetric matrices of order two satisfy together the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion [Formula: see text], uniquely determined up to proper isometries in ℝ3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let [Formula: see text] denote the equivalence class of θ modulo proper isometries in ℝ3 and let [Formula: see text] denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2 (respectively, ≤ 1), have continuous extensions to [Formula: see text], the extension of the field A remaining positive-definite on [Formula: see text], then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to [Formula: see text]. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping ℱ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces [Formula: see text] for the continuous extensions of the matrix fields (A, B), and [Formula: see text] for the continuous extensions of the immersions θ.

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

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