Some uniqueness and continuous dependence theorems for nonlinear elastodynamics in exterior domains

AMS (MOS) Nos. 73C50, 73C15; 35B30, 35M05

Uniqueness and Höolder continuous dependence upon the initial data of the null solution to the initial boundary value problem of nonlinear hyperelasticity are proved for exterior domains subject to mild asymptotic behaviour on the displacement, velocity and stress components. The strain-energy is not required to be locally sign-definite although at sufficiently large spatial distances it must be non-negative. Other limitations imposed on the strain-energy become identically satisfied upon reduction to the linear theory.     

On the continuous dependence in linear anisotropic viscoelasticity on exterior domains

Adoperando il metodo della funzione peso, si dimostrano teoremi di unicità e dipendenza continua per le soluzioni della teoria dinamica della viscoelasticità lineare anisotropa nei domini esterni.     

The continuous dependence on parameters of solutions for a class of elliptic problems on exterior domains

The goal of this paper is to discuss the continuous dependence of solutions on functional parameters for the following semilinear elliptic partial differential equation: , for x∈Ω_r0?{x∈Rⁿ,n≥3,‖x‖>r₀} and v∈V, where V stands for some functional space. Our approach covers the case when f may change sign and admits general growth. As an additional result, the characterization of the radius r₀ for which our problem possesses at least one positive evanescent solution in the exterior domain Ω_r0 is described and numerically illustrated. Our approach relies on the subsolution and supersolution method and on a lemma due to Noussair and Swanson.     

Generalized linear elasticity in exterior domains. I: Radiation problems

We use Hodge–Helmholtz decompositions of weighted Sobolev spaces to solve time-harmonic exterior-boundary value problems for perturbations of the (aδd+bdδ)-system (δ: the co-differential, a, b>0). We prove, that a Fredholm alternative holds true, the eigensolutions decay polynomially at infinity, and that the positive eigenvalues do not accumulate. © 1997 B. G. Teubner Stuttgart-John Wiley & Sons Ltd.     

Large time behavior of anisotropic electromagnetic/elasticity equations in exterior domains

We find uniform rates of decay of the total energy of the coupled system of anisotropic electromagnetic/elasticity model in exterior domains provided mild dissipative effects are present. The decay of the total energy is of polynomial type. The conclusions of this paper improve previous results on the subject.     

Quasilinear elliptic system in exterior domains with dependence on the gradient

In this paper, we deal with a quasilinear elliptic system in exterior domains with dependence on the gradient and coupling of the equations not only inside of the domain, but also on the boundary. We prove the existence of positive, negative or sign changing weak solutions. Our approach relies on an aproximation argument and an adequate elliptic “a priori” estimate.     

Biharmonic problem in exterior domains

We study here a biharmonic equation in an exterior domain of $\text{R}{}^{\mathrm{n}}$. We give in L^p theory, with 1<p<∞ existence, uniqueness and regularity results. To cite this article: C. Amrouche, M. Fontes, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On the first stationary boundary-value problem of elasticity in weighted Sobolev spaces in exterior domains of R3

This paper solves the first boundary-value problem of elasticity both in interior and exterior domains ofR ³. The equations are set in weighted Sobolev spaces for exterior domains that describe the decay of the functions at infinity. The results established include existence, regularity, and convergence of iterations of the solution.This research was supported by the Rashi Foundation.     

Numerical mapping of exterior domains

This paper describes an integral equation method for computing the conformal mapping of the exterior of a given simply-connected region onto the exterior of the unit circle. The method includes evaluation of the transfinite, diameter of the given region. Results are presented for a number of trial problems.     

Pointwise gradient estimates in exterior domains

We consider a class of elliptic operators with unbounded coefficients in a smooth exterior domain Ω and we prove that the Cauchy-Neumann problem associated with admits, for any bounded and continuous initial datum, a unique bounded classical solution. We also provide pointwise gradient estimates for such a solution. Received: 5 July 2005; Revised: 20 December 2005     

The Ornstein-Uhlenbeck semigroup in exterior domains

Let Ω be an exterior domain in It is shown that Ornstein-Uhlenbeck operators L generate C₀-semigroups on L^p(Ω) for p ∈ (1, ∞) provided ∂Ω is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p − L^q smoothing properties of e^tL. If ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2. Received: 16 December 2004; revised: 4 February 2005     

Global existence results for Oldroyd-B fluids in exterior domains

In this paper we consider the set of equations describing Oldroyd-B fluids in exterior domains. It is shown that these equations admit a unique, global solution defined in a certain function space provided the initial data and the coupling constant are small enough.     

Decay estimates for dissipative wave equations in exterior domains

Consider the initial boundary value problem for the linear dissipative wave equation (□+∂_t)u=0 in an exterior domain . Using the so-called cut-off method together with local energy decay and L² decays in the whole space, we study decay estimates of the solutions. In particular, when N?3, we derive L^p decays with p?1 of the solutions. Next, as an application of the decay estimates for the linear equation, we consider the global solvability problem for the semilinear dissipative wave equations (□+∂_t)u=f(u) with f(u)=|u|^α+1,|u|^αu in an exterior domain.     

Uniqueness in exterior domains for the generalized heat conduction

In this note, we prove uniqueness of those solutions of the generalized heat conduction equation that increase as an exponential of the square of the distance from the origin. Continuous dependence results with respect to initial data and supply terms are also proved. The results are obtained with the help of the weighted energy method. We also prove uniqueness of the solutions of the backward in time problem. This second result is obtained by means of the weighted Lagrange identities method.     

Weighted estimates for the incompressible fluid in exterior domains

For a strong solution u(x,t) of the Navier-Stokes equations in exterior domain Ω in Rn where n=2,3, we study the time decay of ‖α|x|u(t)Lp_‖ for α<n. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in [H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier-Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (3) (2007) 547-567; H.-O. Bae, B.J. Jin, Asymptotic behavior for the Navier-Stokes solutions in 2D exterior domains, J. Funct. Anal. 240 (2006) 508-529] and we will extend Bae and Jin's results by modifying their methods.     

Hardy's inequality on exterior domains

Let be a smooth exterior domain in and . We prove that when , Hardy's inequality is valid on .