Showing papers in "Archive for Rational Mechanics and Analysis in 1988"

The effect of a singular perturbation on nonconvex variational problems

Abstract: We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter ɛ → 0. The technique utilizes the notion of “Γ-convergence” of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

Abstract: We prove that viscosity solutions in W1,∞ of the second order, fully nonlinear, equation F(D2u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.

Equilibrium configurations of crystals

Abstract: Unspecified

The Hamiltonian structure of nonlinear elasticity: The material and convective representations of solids, rods, and plates

Abstract: It is our belief that a thorough understanding of the mathematical underpinnings of elasticity is crucial to its analytical and numerical implementation. For example, in the analysis of rotating structures, the coupling of the equations for geometrically inexact models, obtained by linearization or other approximations, with those for rotating rigid bodies can easily lead to misleading artificial “softening” effects that can significantly alter numerical results; see Simo and VuQuoc [1986c] (especially equations (3) and (5)). In this paper, we consider fully nonlinear geometrically exact models for rods, plates (and shells) which take into account shear and torsion as well as the usual bending effects in traditional rod and plate models. These models can be obtained either from the three-dimensional theory by a systematic use of projection methods; see e.g., Antman [1972] and Naghdi [1972], or by a direct approach within the context of Cosserat continuum. Remarkably, the two approaches lead to essentially the same form of the governing field equations. In the present context, we have chosen as a model problem a particular rod model which may be regarded as an extension of the classical Kirchhoff-Love model (see Love [1944]) to include shear deformations, as in

Decay estimates for some semilinear damped hyperbolic problems

Abstract: Let Ω be a bounded open domain in Rn, g ∶ R → R a non-decreasing continuous function such that g(0)=0 and h e Lloc1(R+; L2(Ω)). Under suitable assumptions on g and h, the rate of decay of the difference of two solutions is studied for some abstract evolution equations of the general form u′′ + Lu + g(u′) = h(t,x) as t → + ∞. The results, obtained by use of differential inequalities, can be applied to the case of the semilinear wave equation $$u_u - \Delta u + g{\text{(}}u_t {\text{) = }}h{\text{ in }}R^ + \times \Omega ,{\text{ }}u = {\text{0 on }}R^ + \times \partial \Omega$$ in R+×Ω, u=0 onR+×∂Ω. For instance if \(g(s) = c\left| s \right|^{p - 1} s + d\left| s \right|^{q - 1} s\) with c, d>0 and 1 < p≦q, (n−2)q≦n+2, then if \(h \in L^\infty (R + ;L^2 (\Omega ))\), all solutions are bounded in the energy space for t≧0 and if u, v are two such solutions, the energy norm of u(t) − v(t) decays like t−1/p−1 as t → + ∞.

Regularity properties of deformations with finite energy

Abstract: The purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity.

General existence theorems for unilateral problems in continuum mechanics

Abstract: The problem of minimizing a possibly non-convex and non-coercive functional is studied. Either necessary or sufficient conditions for the existence of solutions are given, involving a generalized recession functional, whose properties are discussed thoroughly. The abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region.

Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law

Abstract: In [1986g, 1988g] I began the development of a nonequilibrium thermomechanics of two-phase continua, a development based on dynamical statements of the thermomechanical laws in conjunction with GIBBS’S notion of a sharp phase-interface endowed with energy and entropy. I have since come to realize that there is an additional balance law appropriate to the interface. This law, which represents balance of capillary forces, has the form1 $$\int\limits_{{\partial c}} {{\text{C}}\upsilon {\text{ + }}\int\limits_{{\text{c}}} \pi } = 0,$$ (7.1) with an arbitrary subsurface of and ν the outward unit normal to the boundary curve ∂ of. Here C (x,t), the capillary stress, is a linear transformation of tangent vectors into (not necessarily tangent) vectors, while π(x,t), the interaction, is a vector field; C (x,t) represents microforces exerted across ∂c in response to the creation of new surface;π(X,t), characterizes the interaction between the interface and the bulk material. I view (1.1) as a balance law which is supplementary to the usual laws for forces and moments.

Exact boundary controllability of a hybrid system of elasticity

Abstract: A hybrid control system is presented: an elastic beam, governed by a partial differential equation, linked to a rigid body which is governed by an ordinary differential equation and to which control forces and torques are applied. The entire system, elastic beam plus rigid body, is proved to be exactly controllable by smooth open-loop controllers applied to the rigid body only, and in arbitrarily short durations. This system is modeled as a two-dimensional space-structure.

On a boundary layer problem for the nonlinear Boltzmann equation

Abstract: This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates.

Homogeneous diffusion in ℝ with power-like nonlinear diffusivity

Abstract: We study the nonnegative solutions of the initial-value problem ut=(ur|ux|p-1ux)x,u(x, 0)∈L1(ℝ), where p>0, r+p>0. The local velocity of propagation of the solutions is identified as V = -vx| vx|p-1 where v =cuα (with r +p - 1)/p and c (r +p/(r +p- 1)) is the nonlinear potential. Our main result is the a priori estimate (vx|vx|p-1)x≥- $$\frac{1}{{(2p + r) t}}$$

Global L p -properties for the spatially homogeneous Boltzmann equation

Abstract: This paper studies the Lp-behavior for 1≦p ≦ ∞ of solutions of the nonlinear, spatially homogeneous Boltzmann equation for a class of collision kernels including inverse kth-power forces with k>5 and angular cut-off. The following topics are treated: differentiability in Lp together with global boundedness in time for Lp-moments that exist initially, translation continuity in Lp uniformly in time, and strong convergence to equilibrium.

Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity

Abstract: Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross-sectional resultants of force and moment. Using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite-dimensional manifold. In particular, the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations. Moreover, the set of semi-inverse solutions can be analyzed locally.

Fit Regions and Functions of Bounded Variation

Abstract: Ever since one of us (W. N.) began the attempt to analyze some of the basic concepts of continuum physics in terms of precise mathematical structures about thirty years ago (see [N1]), it was clear that a concept of “fit region” in a Euclidean space was needed. Only such “fit regions” should be sets fit to be occupied by continuous bodies and their subbodies.

A connection formula for the Second Painlevé Transcendent

Abstract: We consider a particular case of the Second Painleve Transcendent $$y^{''} = xy + 2y^3$$ It is known that if y(x) ∼ kAi(x) as x → +∞, then if 0