Strong Ellipticity Research Articles

Stochastic optimization theory in Hilbert spaces?1

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.

Surface waves in restricted Hadamard materials

The propagation of waves of small amplitude in an elastic medium composed of Hadamard material, satisfying also the Ordered Forces condition and the Strong Ellipticity condition, and subjected to a large homogeneous biaxial stress, is investigated in detail. Consideration is given to the existence and general properties of Rayleigh waves on the surface of a semi-infinite medium, and to the propagation of plate waves in a medium of finite thickness.

Strong ellipticity conditions for the differential operator of the finite isotropic elasticity

TheStrong-Ellipticity conditions of the operator expressing the problem of a state of finite elastic deformation superimposed on a distorted configuration are examined. Sufficient criteria ofLocal Strong-Ellipticity, depending only on the so called Elastic Moduli of the material and on the principal components of stretch in the initial distorted configuration are supplied.

Waves in isotropic elastic materials of Hadamard, Green, or harmonic type

The propagation condition for infinitesimal plane waves superposed on a state of homogeneous strain in isotropic elastic solids is expressed in a simple form in terms of certain instantaneous moduli of elasticity referred to the principal axes of the underlying strain. The constitutive laws of Hadamard, Green, and harmonic materials are obtained with some economy by using the propagation condition in this form. Under certain restrictions these materials are shown to satisfy the strong ellipticity condition.

On the essential spectrum of an elliptic operator perturbed by a potential

In [11] we made a study of the essential spectra of a strongly elliptic operator acting on L2(En). The object was to give sufficient conditions on a potential q to guarantee that the essential spectra of A + q are the same as those of A. The potential q was not assumed to have domain containing the domain of A, but merely to have a dense intersection with it (namely, the set C~ of infinitely differentiably functions with compact support). In this case one must consider some closed extension of A + q acting on C~o (there may be many). We found it convenient to employ regularly accretive extensions in the sense of Kato [13]. Sufficient conditions on q were given in order that a regularly accretive extension of A + q exist. Then further conditions were given in order that this extension have the same essential spectra as A. In this paper we are concerned with the same problem for A acting on LP(E~), 1 < p < oo. In this case strong ellipticity offers no advantage, so we drop this assumption. However, serious difficulties present themselves. On the one hand there is no known theory of closed extensions corresponding to Kato's regularly accretive extensions. On the other, the L 2 theory which is suitable for strongly elliptic operators no longer applies. The first difficulty is surmounted by developing a theory of extensions in an arbitrary Banach space which generalizes Kato's development. We call these operators "intermediate extensions". For a Hilbert space, regularly

Ellipticity and regularity for periodic nonlinear equations

The composition of two strongly elliptic linear operators with suitably differentiable coefficients is also strongly elliptic. This fact has been used [3, pp. 178-181] to yield a particularly simple proof of the regularity of weak solutions of periodic strongly elliptic linear equations. In this paper we prove a similar regularity theorem for periodic nonlinear equations under the assumption of strong ellipticity for the composition of the given nonlinear operator with certain linear operators. Let A be a nonlinear differential operator of order 2m given by

PNI-equations

IN this paper, on the basis of some integral identities for the solutions of multi-group kinetic equations an algorithm is constructed for forming sets of partial differential equations in coordinate space for solving kinetic problems approximately. We call the equations so obtained P NI -equations (see [1], [2]), they can be used both in P-operations in the KP-method [1], [2], and also to construct finite difference analogs of the kinetic equation. The method by which the P NI -equations will be obtained can be called the generalized Galerkin method (see [3]); the P NI -equations for some problems of transport theory are a generalization of such methods, e.g. the method of spherical harmonics [4]-[6], the KP-method [7], the method of characteristics [8], [9], Evans' method [4], and the method of decomposition of the solution into Jacobian polynomials [1O]. With regard to the last method this paper indicates the necessity of a correspondence between the dimensionality of the space in which the problem is solved and the representative Jacobian polynomial. In the choice of the form of the P NI -equations there is a fair amount of arbitrariness; in particular we can confine ourselves to considering strongly elliptic sets of equations, although up to now the strong ellipticity of the equations of the method of spherical harmonics has not yet been proved. It is also known that the working out of numerical algorithms for the calculation of multi-dimensional transport problems has been retarded because of the absence of sufficiently good quadrature formulae for a sphere, which may be used for a good approximation of the kinetic equation with as dense a network as we please with respect to spatial and angular variables, in PWJ-equations this difficulty is in some measure removed; with the choice of the best divisions with regard to the angle, we can use the results of paper [11] for some problems. In the choice of P NI -equations we shall confine ourselves to considering the case with an isotropic dispersion indicatrix, although the algorithm for derivation put forward can obviously be extended to the case of a degenerate indicatrix. In this paper, on the basis of the P NI -equations, various finite difference analogs for the kinetic equations are obtained; the convergence is proved in corresponding spaces of solutions of P NI -equations and of their difference analogs to the solution of the kinetic equation.

Wave propagation and uniqueness in prestressed elastic solids

A homogeneous isotropic perfectly elastic body is subjected to a large static pure homogeneous deformation with two of the principal extension ratios equal. An infinitesimal deformation is superimposed on the large deformation. The conditions for strong ellipticity of the system of equilibrium equations for the infinitesimal deformation are obtained. These conditions are examined within the context of uniqueness or non-uniqueness of the displacement boundary-value problem for the infinitesimal deformation. It is found that the conditions of strong ellipticity are sufficient but not necessary for uniqueness.