Journal of functional analysis, ISSN 0022-1236, 2014, Volume 266, Issue 7, pp. 4314 - 4421

...) is defined as the completion in the classical Sobolev space Wk,p(O) of (restrictions to O of) functions from Cc∞(Rn) whose supports are disjoint from D...

Bessel potential space and capacity | Synthesis | Locally [formula omitted]-domain | Mixed boundary value problem | Higher-order elliptic system | Ahlfors regular set | Linear extension operator | Besov and Triebel–Lizorkin spaces | Higher-order Sobolev space | Higher-order boundary trace operator | Real and complex interpolation | Locally (ε, δ)-domain | Besov and Triebel-Lizorkin spaces | DIFFERENTIABLE FUNCTIONS | Locally (epsilon, delta)-domain | STOKES SYSTEM | BESOV-SPACES | LIPSCHITZ-DOMAINS | EXTENSION-THEOREMS | INTERPOLATION | MATHEMATICS | DECOMPOSITIONS | REGULARITY | DIRICHLET PROBLEM | ELLIPTIC-EQUATIONS

Bessel potential space and capacity | Synthesis | Locally [formula omitted]-domain | Mixed boundary value problem | Higher-order elliptic system | Ahlfors regular set | Linear extension operator | Besov and Triebel–Lizorkin spaces | Higher-order Sobolev space | Higher-order boundary trace operator | Real and complex interpolation | Locally (ε, δ)-domain | Besov and Triebel-Lizorkin spaces | DIFFERENTIABLE FUNCTIONS | Locally (epsilon, delta)-domain | STOKES SYSTEM | BESOV-SPACES | LIPSCHITZ-DOMAINS | EXTENSION-THEOREMS | INTERPOLATION | MATHEMATICS | DECOMPOSITIONS | REGULARITY | DIRICHLET PROBLEM | ELLIPTIC-EQUATIONS

Journal Article

Communications on Pure and Applied Mathematics, ISSN 0010-3640, 08/2014, Volume 67, Issue 8, pp. 1219 - 1262

For a family of second‐order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors...

MATHEMATICS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | DIRICHLET PROBLEM | LIPSCHITZ-DOMAINS | DIVERGENCE FORM | EQUATION | SINGULAR-INTEGRALS | COMPACTNESS METHODS

MATHEMATICS | MATHEMATICS, APPLIED | BOUNDARY-VALUE-PROBLEMS | DIRICHLET PROBLEM | LIPSCHITZ-DOMAINS | DIVERGENCE FORM | EQUATION | SINGULAR-INTEGRALS | COMPACTNESS METHODS

Journal Article

Communications in nonlinear science & numerical simulation, ISSN 1007-5704, 2015, Volume 21, Issue 1-3, pp. 99 - 111

...’ algorithm is proposed in its basic version.•The method is based on efficient diagonal partitions and smooth auxiliary functions...

Lipschitz gradients | Deterministic methods | Global optimization | Unknown Lipschitz constant | MATHEMATICS, APPLIED | WORKING | SET | SEARCH | PHYSICS, FLUIDS & PLASMAS | INFORMATION | PHYSICS, MATHEMATICAL | SOFTWARE | STRATEGIES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | LIPSCHITZ-CONSTANTS | 1ST DERIVATIVES | MINIMIZATION ALGORITHM | PARTITION | Lipschitz condition | Algorithms | Mathematical analysis | Constants | Mathematical models | Representations | Optimization | Convergence

Lipschitz gradients | Deterministic methods | Global optimization | Unknown Lipschitz constant | MATHEMATICS, APPLIED | WORKING | SET | SEARCH | PHYSICS, FLUIDS & PLASMAS | INFORMATION | PHYSICS, MATHEMATICAL | SOFTWARE | STRATEGIES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | LIPSCHITZ-CONSTANTS | 1ST DERIVATIVES | MINIMIZATION ALGORITHM | PARTITION | Lipschitz condition | Algorithms | Mathematical analysis | Constants | Mathematical models | Representations | Optimization | Convergence

Journal Article

Advances in mathematics (New York. 1965), ISSN 0001-8708, 2010, Volume 224, Issue 3, pp. 910 - 966

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz...

Operator Hölder functions | Multiple operator integrals | Contractions | Hölder classes | Zygmund class | Unitary operators | Operator Lipschitz function | Self-adjoint operators | Holder classes | INTEGRALS | MATHEMATICS | Operator Holder functions | LIPSCHITZ

Operator Hölder functions | Multiple operator integrals | Contractions | Hölder classes | Zygmund class | Unitary operators | Operator Lipschitz function | Self-adjoint operators | Holder classes | INTEGRALS | MATHEMATICS | Operator Holder functions | LIPSCHITZ

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 2019, Volume 266, Issue 1, pp. 44 - 69

.... We study extensions of Newtonian Sobolev functions to the completion Xˆ of X and use them to obtain several results on X itself, in particular concerning minimal weak...

Newtonian Sobolev space | Lebesgue point | Quasiminimizer | Noncomplete metric space | p-harmonic function | Poincaré inequality | QUASICONTINUITY | QUASIMINIMIZERS | LIPSCHITZ FUNCTIONS | POTENTIAL-THEORY | OPEN SETS | MATHEMATICS | QUASIOPEN | Poincare inequality | HARMONIC-FUNCTIONS

Newtonian Sobolev space | Lebesgue point | Quasiminimizer | Noncomplete metric space | p-harmonic function | Poincaré inequality | QUASICONTINUITY | QUASIMINIMIZERS | LIPSCHITZ FUNCTIONS | POTENTIAL-THEORY | OPEN SETS | MATHEMATICS | QUASIOPEN | Poincare inequality | HARMONIC-FUNCTIONS

Journal Article

Acta mathematica, ISSN 0001-5962, 2011, Volume 207, Issue 2, pp. 375 - 389

.... In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S
α
, 1 < α...

Mathematics, general | Mathematics | Schatten–von Neumann ideals | Operator-Lipschitz functions | Schatten-von Neumann ideals | INTEGRALS | MATHEMATICS | CONTINUITY | SPACES | Perturbation theory | Norms | Errors | Syntax | Mathematical analysis | Linear operators

Mathematics, general | Mathematics | Schatten–von Neumann ideals | Operator-Lipschitz functions | Schatten-von Neumann ideals | INTEGRALS | MATHEMATICS | CONTINUITY | SPACES | Perturbation theory | Norms | Errors | Syntax | Mathematical analysis | Linear operators

Journal Article

Journal of optimization theory and applications, ISSN 1573-2878, 2019, Volume 182, Issue 3, pp. 885 - 905

Given an open subset of a Banach space and a Lipschitz real-valued function defined on its closure, we study whether it is possible to approximate this function uniformly by Lipschitz functions having...

54C30 | 58C25 | Mathematics | Theory of Computation | Optimization | Smooth approximation | Calculus of Variations and Optimal Control; Optimization | 41A30 | Operations Research/Decision Theory | Almost classical solution | 41A65 | Eikonal equation | 26A16 | Lipschitz function | 26B05 | Applications of Mathematics | Engineering, general | 41A29 | SPACE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EXTENSION | REGULARIZATION | Banach space | Mathematical functions | Functional Analysis

54C30 | 58C25 | Mathematics | Theory of Computation | Optimization | Smooth approximation | Calculus of Variations and Optimal Control; Optimization | 41A30 | Operations Research/Decision Theory | Almost classical solution | 41A65 | Eikonal equation | 26A16 | Lipschitz function | 26B05 | Applications of Mathematics | Engineering, general | 41A29 | SPACE | MATHEMATICS, APPLIED | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | EXTENSION | REGULARIZATION | Banach space | Mathematical functions | Functional Analysis

Journal Article

Advances in mathematics (New York. 1965), ISSN 0001-8708, 06/2019, Volume 349, pp. 1198 - 1233

..., where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space...

Observable variance | Concentration of measure | Lipschitz function | Isoperimetric profile | Metric measure space | METRIC-MEASURE-SPACES | INEQUALITIES | RICCI CURVATURE | MATHEMATICS | THEOREMS | SHARP | GEOMETRY | SURFACES

Observable variance | Concentration of measure | Lipschitz function | Isoperimetric profile | Metric measure space | METRIC-MEASURE-SPACES | INEQUALITIES | RICCI CURVATURE | MATHEMATICS | THEOREMS | SHARP | GEOMETRY | SURFACES

Journal Article

2014, De Gruyter studies in mathematics, ISBN 9783110281231, Volume 52., xiii, 449

Book

Computers & mathematics with applications (1987), ISSN 0898-1221, 2010, Volume 60, Issue 10, pp. 2779 - 2787

The goal of this paper is to unify and extend the generating functions of the generalized Bernoulli polynomials, the generalized Euler polynomials and the generalized Genocchi polynomials associated...

Riemann and Hurwitz (or generalized) zeta functions | Polylogarithm function | Lipschitz–Lerch zeta function | Euler numbers and Euler polynomials | Lerch zeta function | Bernoulli numbers and Bernoulli polynomials | Genocchi numbers and Genocchi polynomials | Recurrence relations | Hurwitz–Lerch zeta function | Mellin transformation | Dirichlet character | LipschitzLerch zeta function | HurwitzLerch zeta function | MATHEMATICS, APPLIED | Hurwitz-Lerch zeta function | NUMBERS | EXTENSION | Lipschitz-Lerch zeta function | ZETA | APOSTOL-BERNOULLI | FORMULAS | Genocchi numbers and Genocch polynomials | Construction | Mathematical models | Transformations | Mathematical analysis

Riemann and Hurwitz (or generalized) zeta functions | Polylogarithm function | Lipschitz–Lerch zeta function | Euler numbers and Euler polynomials | Lerch zeta function | Bernoulli numbers and Bernoulli polynomials | Genocchi numbers and Genocchi polynomials | Recurrence relations | Hurwitz–Lerch zeta function | Mellin transformation | Dirichlet character | LipschitzLerch zeta function | HurwitzLerch zeta function | MATHEMATICS, APPLIED | Hurwitz-Lerch zeta function | NUMBERS | EXTENSION | Lipschitz-Lerch zeta function | ZETA | APOSTOL-BERNOULLI | FORMULAS | Genocchi numbers and Genocch polynomials | Construction | Mathematical models | Transformations | Mathematical analysis

Journal Article

Mathematical Programming, ISSN 0025-5610, 6/2012, Volume 133, Issue 1, pp. 299 - 325

... of unsuccessful iterates, if the function being minimized is Lipschitz continuous near the limit point...

Discontinuity | Lower semicontinuity | Theoretical, Mathematical and Computational Physics | Nonsmooth calculus | Mathematics | Direct-search methods | Mathematical Methods in Physics | Generalized directional derivatives | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | Numerical Analysis | 90C56 | Combinatorics | Lipschitz extensions | Directionally Lipschitz | MATHEMATICS, APPLIED | ALGORITHMS | ADAPTIVE DIRECT SEARCH | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | OPTIMIZATION | GRADIENTS | Studies | Analysis | Mathematical programming | Constraints | Mathematical analysis | Searching | Mathematical models | Derivatives | Optimization | Step functions

Discontinuity | Lower semicontinuity | Theoretical, Mathematical and Computational Physics | Nonsmooth calculus | Mathematics | Direct-search methods | Mathematical Methods in Physics | Generalized directional derivatives | 90C30 | Mathematics of Computing | Calculus of Variations and Optimal Control; Optimization | Numerical Analysis | 90C56 | Combinatorics | Lipschitz extensions | Directionally Lipschitz | MATHEMATICS, APPLIED | ALGORITHMS | ADAPTIVE DIRECT SEARCH | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | CONVERGENCE | OPTIMIZATION | GRADIENTS | Studies | Analysis | Mathematical programming | Constraints | Mathematical analysis | Searching | Mathematical models | Derivatives | Optimization | Step functions

Journal Article

2019, Lecture Notes in Mathematics, ISBN 9783030164881, Volume 2241, 605

eBook

Advances in computational mathematics, ISSN 1572-9044, 2015, Volume 42, Issue 2, pp. 333 - 360

This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds...

58C05 | Visualization | Computational Mathematics and Numerical Analysis | Riemannian manifolds | 65K05 | Mathematical and Computational Biology | Mathematics | Computational Science and Engineering | Descent direction | Clarke subdifferential | 49J52 | Lipschitz function | Mathematical Modeling and Industrial Mathematics | MATHEMATICS, APPLIED | STEEPEST DESCENT METHOD | NEWTON METHOD | SUBSETS | OPTIMIZATION | NONSMOOTH | COMPLETION | CONVEX-FUNCTIONS

58C05 | Visualization | Computational Mathematics and Numerical Analysis | Riemannian manifolds | 65K05 | Mathematical and Computational Biology | Mathematics | Computational Science and Engineering | Descent direction | Clarke subdifferential | 49J52 | Lipschitz function | Mathematical Modeling and Industrial Mathematics | MATHEMATICS, APPLIED | STEEPEST DESCENT METHOD | NEWTON METHOD | SUBSETS | OPTIMIZATION | NONSMOOTH | COMPLETION | CONVEX-FUNCTIONS

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 2018, Volume 464, Issue 1, pp. 473 - 492

We study the Daugavet property in the space of Lipschitz functions Lip0(M) on a complete metric space M...

Strongly exposed point | Daugavet property | Space of Lipschitz functions | Lipschitz-free space | Length space | MATHEMATICS | MATHEMATICS, APPLIED

Strongly exposed point | Daugavet property | Space of Lipschitz functions | Lipschitz-free space | Length space | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

Journal of functional analysis, ISSN 0022-1236, 2018, Volume 275, Issue 8, pp. 2015 - 2058

.... Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields...

Bi-Lipschitz maps | Littlewood–Paley diagonalization | Directional Hilbert transform | Directional square functions | MATHEMATICS | FIELDS | PLANE | HILBERT-TRANSFORMS | Littlewood-Paley diagonalization | SINGULAR-INTEGRALS | Mathematics - Classical Analysis and ODEs

Bi-Lipschitz maps | Littlewood–Paley diagonalization | Directional Hilbert transform | Directional square functions | MATHEMATICS | FIELDS | PLANE | HILBERT-TRANSFORMS | Littlewood-Paley diagonalization | SINGULAR-INTEGRALS | Mathematics - Classical Analysis and ODEs

Journal Article

2012, Annals of mathematics studies, ISBN 9780691153551, Volume no. 179, ix, 425

...-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces...

Calculus of variations | Functional analysis | Banach spaces | Mathematics | Frechet spaces | Lipschitz spaces

Calculus of variations | Functional analysis | Banach spaces | Mathematics | Frechet spaces | Lipschitz spaces

Book

Mathematical Programming, ISSN 0025-5610, 9/2018, Volume 171, Issue 1, pp. 463 - 487

In this paper we study local error bound moduli for a locally Lipschitz and regular function via outer limiting subdifferential sets...

Outer limiting subdifferential | Theoretical, Mathematical and Computational Physics | End set | Mathematics | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Locally Lipschitz | Numerical Analysis | 90C34 | Lower $${\mathcal {C}}^1$$ C 1 function | Support function | 65K10 | Combinatorics | Error bound modulus | Lower C | function | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | CALMNESS | WEAK SHARP MINIMA | SUFFICIENT CONDITIONS | STABILITY | Lower C-1 function | CONSTRAINT QUALIFICATIONS | CONVEX INEQUALITIES | METRIC REGULARITY | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BANACH-SPACES | LINEAR INEQUALITY SYSTEMS | Error analysis | Constraining

Outer limiting subdifferential | Theoretical, Mathematical and Computational Physics | End set | Mathematics | Mathematical Methods in Physics | 90C30 | Calculus of Variations and Optimal Control; Optimization | Mathematics of Computing | Locally Lipschitz | Numerical Analysis | 90C34 | Lower $${\mathcal {C}}^1$$ C 1 function | Support function | 65K10 | Combinatorics | Error bound modulus | Lower C | function | LOWER SEMICONTINUOUS FUNCTIONS | MATHEMATICS, APPLIED | CALMNESS | WEAK SHARP MINIMA | SUFFICIENT CONDITIONS | STABILITY | Lower C-1 function | CONSTRAINT QUALIFICATIONS | CONVEX INEQUALITIES | METRIC REGULARITY | COMPUTER SCIENCE, SOFTWARE ENGINEERING | OPERATIONS RESEARCH & MANAGEMENT SCIENCE | BANACH-SPACES | LINEAR INEQUALITY SYSTEMS | Error analysis | Constraining

Journal Article

SIAM journal on optimization, ISSN 1095-7189, 2013, Volume 23, Issue 1, pp. 508 - 529

...: (i) the objective function f(x) satisfies the Lipschitz condition with a constant L; (ii) the first derivative of f(x...

Lipschitz functions | Global optimization | Balancing local and global information | Acceleration | Lipschitz derivatives | balancing local and global information | acceleration | MATHEMATICS, APPLIED | MINIMAL ROOT | SET | global optimization | Construction | Lipschitz condition | Algorithms | Mathematical analysis | Searching | Mathematical models | Derivatives | Optimization

Lipschitz functions | Global optimization | Balancing local and global information | Acceleration | Lipschitz derivatives | balancing local and global information | acceleration | MATHEMATICS, APPLIED | MINIMAL ROOT | SET | global optimization | Construction | Lipschitz condition | Algorithms | Mathematical analysis | Searching | Mathematical models | Derivatives | Optimization

Journal Article

Journal de mathématiques pures et appliquées, ISSN 0021-7824, 12/2019, Volume 132, pp. 457 - 482

Given an arbitrary planar ∞-harmonic function u, for each α>0 we establish a quantitative Wloc1,2-estimate of |Du|α...

Sobolev regularity | harmonic function | Absolute minimizer | LAPLACIAN | MATHEMATICS | MATHEMATICS, APPLIED | infinity-harmonic function | MINIMIZATION PROBLEMS | LIPSCHITZ EXTENSIONS

Sobolev regularity | harmonic function | Absolute minimizer | LAPLACIAN | MATHEMATICS | MATHEMATICS, APPLIED | infinity-harmonic function | MINIMIZATION PROBLEMS | LIPSCHITZ EXTENSIONS

Journal Article