Plucinsky, Paul Lemm, Marius Bhattacharya, Kaushik
Published in
Archive for Rational Mechanics and Analysis

Nematic elastomers and glasses deform spontaneously when subjected to temperature changes. This property can be exploited in the design of heterogeneously patterned thin sheets that deform into a non-trivial shape when heated or cooled. In this paper, we start from a variational formulation for the entropic elastic energy of liquid crystal elastome...

Frid, Hermano Li, Yachun
Published in
Archive for Rational Mechanics and Analysis

We consider a mixed type boundary value problem for a class of degenerate parabolic–hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for L2-divergence-measur...

Le Floch, Bruno LeFloch, Philippe G.
Published in
Archive for Rational Mechanics and Analysis

We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in ...

Bernardin, C. Gonçalves, P. Jiménez-Oviedo, B.
Published in
Archive for Rational Mechanics and Analysis

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy...

Kholmatov, Shokhrukh Yu. Piovano, Paolo

A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wid...

Caffarelli, L. Cagnetti, F. Figalli, A.
Published in
Archive for Rational Mechanics and Analysis

We study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Under smoothness assumptions on the boundary conditions and on the fracture energy density, we show that minimizers are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfo...

Armstrong, Scott Ferguson, Samuel J Kuusi, Tuomo
Published in
Archive for rational mechanics and analysis

We prove large-scale C ∞ regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian L ¯ , (ii) the...

Colombo, Maria Mingione, Giuseppe
Published in
Archive for Rational Mechanics and Analysis

We prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases, according to the zero set of a modulating coefficient a(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepac...

Colombo, Maria Mingione, Giuseppe
Published in
Archive for Rational Mechanics and Analysis

Bounded minimisers of the functional w↦∫(|Dw|p+a(x)|Dw|q)dx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w \mapsto \int (|Dw|^p+a(x)|Dw|^q)\,{\rm d}x,$$\end{document...

Ambrosio, Luigi Colombo, Maria Figalli, Alessio
Published in
Archive for Rational Mechanics and Analysis

In this paper we provide a complete analogy between the Cauchy–Lipschitz and the DiPerna–Lions theories for ODE’s, by developing a local version of the DiPerna–Lions theory. More precisely, we prove the existence and uniqueness of a maximal regular flow for the DiPerna–Lions theory using only local regularity and summability assumptions on the vect...