13 results match your criteria Calculus Of Variations And Partial Differential Equations[Journal]

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On the geometry of geodesics in discrete optimal transport.

Calc Var Partial Differ Equ 2019 11;58(1):19. Epub 2018 Dec 11.

3Institute of Mathematics, Friedrich Schiller University Jena, 07737 Jena, Germany.

We consider the space of probability measures on a discrete set , endowed with a dynamical optimal transport metric. Given two probability measures supported in a subset , it is natural to ask whether they can be connected by a constant speed geodesic with support in at all times. Our main result answers this question affirmatively, under a suitable geometric condition on introduced in this paper. Read More

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http://dx.doi.org/10.1007/s00526-018-1456-1DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6390900PMC
December 2018
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Ground states in the diffusion-dominated regime.

Calc Var Partial Differ Equ 2018 11;57(5):127. Epub 2018 Aug 11.

4Dipartimento di Ingegneria, Università degli Studi di Napoli "Parthenope", 80143 Naples, Italy.

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller-Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. Read More

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http://link.springer.com/10.1007/s00526-018-1402-2
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http://dx.doi.org/10.1007/s00526-018-1402-2DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6190998PMC
August 2018
1 Read

Global existence of Dirac-wave maps with curvature term on expanding spacetimes.

Calc Var Partial Differ Equ 2018 20;57(5):119. Epub 2018 Jul 20.

2Department of Mathematics, University of Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany.

We prove the global existence of Dirac-wave maps with curvature term with small initial data on globally hyperbolic manifolds of arbitrary dimension which satisfy a suitable growth condition. In addition, we also prove a global existence result for wave maps under similar assumptions. Read More

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http://dx.doi.org/10.1007/s00526-018-1389-8DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6190925PMC

Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media.

Authors:
Shane Cooper

Calc Var Partial Differ Equ 2018 27;57(3):76. Epub 2018 Apr 27.

Department of Mathematical Sciences, Durham University, Lower Mountjoy, Durham, DH1 3LE UK.

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Read More

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http://dx.doi.org/10.1007/s00526-018-1365-3DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6435209PMC
April 2018
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Horizontal curves of hyperbolic metrics.

Calc Var Partial Differ Equ 2018 15;57(4):106. Epub 2018 Jun 15.

2Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

We analyse the fine convergence properties of one parameter families of hyperbolic metrics that move always in a horizontal direction, i.e. orthogonal to the action of diffeomorphisms. Read More

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http://dx.doi.org/10.1007/s00526-018-1386-yDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6432971PMC
June 2018
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Minkowski valuations on convex functions.

Calc Var Partial Differ Equ 2017 23;56(6):162. Epub 2017 Oct 23.

Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/1046, 1040 Wien, Austria.

A classification of [Formula: see text] contravariant Minkowski valuations on convex functions and a characterization of the projection body operator are established. The associated LYZ measure is characterized. In addition, a new [Formula: see text] covariant Minkowski valuation on convex functions is defined and characterized. Read More

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http://dx.doi.org/10.1007/s00526-017-1243-4DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5653878PMC
October 2017
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Uniaxial versus biaxial character of nematic equilibria in three dimensions.

Calc Var Partial Differ Equ 2017 4;56(2):55. Epub 2017 Apr 4.

2Dipartimento di Matematica "G. Castelnuovo", Sapienza Università di Roma, 00185 Rome, Italy.

We study global minimizers of the Landau-de Gennes (LdG) energy functional for nematic liquid crystals, on arbitrary three-dimensional simply connected geometries with topologically non-trivial and physically relevant Dirichlet boundary conditions. Our results are specific to an asymptotic limit coined in terms of a dimensionless temperature and material-dependent parameter, and some constraints on the material parameters, and we work in the limit that captures features of the widely used Lyuksyutov constraint (Kralj and Virga in J Phys A 34:829-838, 2001). We prove (i) that (re-scaled) global LdG minimizers converge uniformly to a (minimizing) limiting harmonic map, away from the singular set of the limiting map; (ii) we have points of maximal biaxiality and uniaxiality near each singular point of the limiting map; (iii) estimates for the size of "strongly biaxial" regions in terms of the parameter . Read More

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http://dx.doi.org/10.1007/s00526-017-1142-8DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7010394PMC

Stable spike clusters for the precursor Gierer-Meinhardt system in .

Calc Var Partial Differ Equ 2017 22;56(5):142. Epub 2017 Sep 22.

3Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong.

We consider the Gierer-Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer we construct a spike cluster consisting of spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. Read More

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http://dx.doi.org/10.1007/s00526-017-1233-6DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6961491PMC
September 2017

Variational approach to coarse-graining of generalized gradient flows.

Calc Var Partial Differ Equ 2017 28;56(4):100. Epub 2017 Jun 28.

4CERMICS, Ecole des Ponts ParisTech, Champs sur Marne, France.

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with non-dissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. Read More

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http://dx.doi.org/10.1007/s00526-017-1186-9DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6560519PMC
June 2017
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Estimates for capacity and discrepancy of convex surfaces in sieve-like domains with an application to homogenization.

Calc Var Partial Differ Equ 2016 2;55(6):138. Epub 2016 Nov 2.

2Department of Mathematics, Uppsala University, Uppsala, Sweden.

We consider the intersection of a convex surface with a periodic perforation of , which looks like a sieve, given by where is a given compact set and is the size of the perforation in the -cell . When tends to zero we establish uniform estimates for -capacity, , of the set . Additionally, we prove that the intersections are uniformly distributed over and give estimates for the discrepancy of the distribution. Read More

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http://dx.doi.org/10.1007/s00526-016-1088-2DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7175654PMC
November 2016

Refined asymptotics of the Teichmüller harmonic map flow into general targets.

Calc Var Partial Differ Equ 2016 27;55(4):85. Epub 2016 Jun 27.

1Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

The Teichmüller harmonic map flow is a gradient flow for the harmonic map energy of maps from a closed surface to a general closed Riemannian target manifold of any dimension, where both the map and the domain metric are allowed to evolve. Given a weak solution of the flow that exists for all time , we find a sequence of times at which the flow at different scales converges to a collection of branched minimal immersions with no loss of energy. We do this by developing a compactness theory, establishing no loss of energy, for sequences of almost-minimal maps. Read More

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http://dx.doi.org/10.1007/s00526-016-1019-2DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7175677PMC

Infinite-body optimal transport with Coulomb cost.

Calc Var Partial Differ Equ 2015 17;54(1):717-742. Epub 2014 Dec 17.

3Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada.

We introduce and analyze symmetric infinite-body optimal transport (OT) problems with cost function of pair potential form. We show that for a natural class of such costs, the optimizer is given by the independent product measure all of whose factors are given by the one-body marginal. This is in striking contrast to standard finite-body OT problems, in which the optimizers are typically highly correlated, as well as to infinite-body OT problems with Gangbo-Swiech cost. Read More

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http://dx.doi.org/10.1007/s00526-014-0803-0DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7010387PMC
December 2014

On the structure of solutions to a class of quasilinear elliptic Neumann problems. Part II.

Authors:
Chunshan Zhao Yi Li

Calc Var Partial Differ Equ 2005 May;212(1):208-233

Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460, USA.

We continue our work (Y. Li, C. Zhao in J Differ Equ 212:208-233, 2005) to study the structure of positive solutions to the equation epsilon(m) Delta(m)u - u(m-1) + f(u) = 0 with homogeneous Neumann boundary condition in a smooth bounded domain of RN (N >/= 2). Read More

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http://dx.doi.org/10.1016/j.jde.2004.07.021DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2917850PMC
May 2005
3 Reads
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