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Arch. *Rational* *Mech*. Anal. Digital Object Identifier (DOI) 10.1007/s00205-012-0552-1

"... We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, w ..."

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We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, we construct a fivedimensional continuum of T3s and show that its intersection with the boundary of the symmetrised rank-one convex hull is four-dimensional. We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, as far as we know, has not been experimentally observed. Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains. 1.

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Arch. *Rational* *Mech*. Anal. Digital Object Identifier (DOI) 10.1007/s00205-008-0159-8 Gradient Estimates for the Perfect Conductivity Problem

"... We establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect (stiff) conductors are closely spaced inside an open bounded domain and away from the boundary. These results give the optimal blow-up rates of the stress for conductors ..."

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We establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect (stiff) conductors are closely spaced inside an open bounded domain and away from the boundary. These results give the optimal blow-up rates of the stress for conductors with arbitrary shape and in all dimensions. 1.

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Digital Object Identifier (DOI) 10.1007/s00205-007-0080-6 Arch. *Rational* *Mech*. Anal. Crack Initiation in Brittle Materials

"... In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic mater ..."

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In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time ti> 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional

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Arch. *Rational* *Mech*. Anal. 207 (2013) 39–74 Digital Object Identifier (DOI) 10.1007/s00205-012-0552-1

"... We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, w ..."

Abstract
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We study the symmetrised rank-one convex hull of monoclinic-I martensite (a twelve-variant material) in the context of geometrically-linear elasticity. We construct sets of T3s, which are (non-trivial) symmetrised rank-one convex hulls of three-tuples of pairwise incompatible strains. In addition, we construct a fivedimensional continuum of T3s and show that its intersection with the boundary of the symmetrised rank-one convex hull is four-dimensional. We also show that there is another kind of monoclinic-I martensite with qualitatively different semi-convex hulls which, as far as we know, has not been experimentally observed. Our strategy is to combine understanding of the algebraic structure of symmetrised rank-one convex cones with knowledge of the faceting structure of the convex polytope formed by the strains. 1.

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Arch. *Rational* *Mech*. Anal. (2006) Digital Object Identifier (DOI) 10.1007/s00205-006-0011-y Self-Contact for Rods on Cylinders

"... We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the ..."

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We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods. 1.

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Appeared in Arch *Ration* *Mech* Anal 154(2000)1:3-51. LINEAR STABILITY OF STEADY STATES FOR THIN FILM AND CAHN{HILLIARD TYPE EQUATIONS

"... Abstract. We study the linear stability of smooth steady states of the evolution equation ht = −(f(h)hxxx)x − (g(h)hx)x − ah under both periodic and Neumann boundary conditions. If a 6 = 0 we assume f 1. In particular we consider positive periodic steady states of thin lm equations, where a = 0 and ..."

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Abstract. We study the linear stability of smooth steady states of the evolution equation ht = −(f(h)hxxx)x − (g(h)hx)x − ah under both periodic and Neumann boundary conditions. If a 6 = 0 we assume f 1. In particular we consider positive periodic steady states of thin lm equations, where a = 0 and f; g might have degeneracies such as f(0) = 0 as well as singularities like g(0) = +1. If a 0, we prove each periodic steady state is linearly unstable with respect to volume (area) preserving perturbations whose period is an integer multiple of the steady state’s period. For area preserving perturbations having the same period as the steady state, we prove linear instability for all a if the ratio g=f is a convex function. Analogous results hold for Neumann boundary conditions. The rest of the paper concerns the special case of a = 0 and power law coecients f(y) = yn and g(y) = Bym. We characterize the linear stability of each positive periodic steady state under perturbations of the same period. For steady states that do not have a linearly unstable direction, we nd all neutral directions. Surprisingly, our instability results imply a nonexistence result: for a large range of exponents m and n there cannot be two positive periodic steady states with the same period and volume.

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Arch. *Rational* *Mech*. Anal. (2006) Digital Object Identifier (DOI) 10.1007/s00205-006-0029-1 Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

"... The Nosé–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according ..."

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The Nosé–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nosé–Hoover dynamics are nonergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé–Hoover thermostat does not give an ergodynamical system for the onedimensional harmonic oscillator when the “mass ” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass. 1.

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Digital Object Identifier (DOI) 10.1007/s00205-010-0327-5 Arch. *Rational* *Mech*. Anal. Analysis of a Stochastic Implicit Interface Model for an Immersed Elastic Surface in a Fluctuating Fluid

"... We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible vis-cous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the well-posedness of pathwise solutions and provide ..."

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We present some mathematical analyses of a recently proposed stochastic implicit interface model for an elastic surface immersed in an incompressible vis-cous fluid subject to fluctuation forces. We derive suitable a priori estimates and establish the well-posedness of pathwise solutions and provide uniform control on the solutions in probability. 1.

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Arch. *Rational* *Mech*. Anal. 174 (2004) 365–384 Digital Object Identifier (DOI) 10.1007/s00205-004-0331-8 Asymptotic States of a Smoluchowski Equation

"... We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. 1. ..."

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We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers. 1.

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Arch. *Rational* *Mech*. Anal. 192 (2009) 375–401 Digital Object Identifier (DOI) 10.1007/s00205-008-0149-x Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

"... We study the solutions of the nonstationary incompressible Navier–Stokes equations in Rd, d ≧ 2, of self-similar form u(x, t) = 1 () √ x√t U, obtained from t small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity f ..."

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We study the solutions of the nonstationary incompressible Navier–Stokes equations in Rd, d ≧ 2, of self-similar form u(x, t) = 1 () √ x√t U, obtained from t small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x).