Riesz Schauder Theory Research Articles

Shape optimization in acoustic–structure interaction

PurposeMotivated by the acoustics of motor vehicles, a coupled fluid–solid system is considered. The air pressure is modeled by the Helmholtz equation, and the structure displacement is described by elastodynamic equations. The acoustic–structure interaction is modeled by coupling conditions on the common interface. First, the existence and uniqueness of solutions are investigated, and then, after recalling fundamental notions of shape optimization, the tensor form of the distributed shape derivative is obtained for the coupled problem. It is then applied to the minimization of the sound pressure by variation of the structure shape through the positioning of beads.Design/methodology/approachThe existence and uniqueness of solutions up to eigenfrequencies are shown by the Fredholm–Riesz–Schauder theory using a novel decomposition into an isomorphism and a compact operator. For the design optimization, the distributed shape derivative is obtained using the averaged adjoint method. It is then used in a closed 3D optimization process of the position of a bead for noise reduction. In this process, the C++ library concepts are used to solve the differential equations on hexahedral meshes with the finite element method of higher order.FindingsThe existence and uniqueness of solutions have been shown for the case without absorption, where the given proof allows for extension to the case with absorption in the domain or via boundary conditions. The theoretical results show that the averaged adjoint can be applied to compute distributed shape derivatives in the context of acoustic–structure interaction. The numerical results show that the distributed shape derivative can be used to reduce the sound pressure at a chosen frequency via rigid motions of a nonsmooth shape.Originality/valueThe proof of shape differentiability and the calculation of the distributed shape derivative in tensor form allows to consider nonsmooth shapes for the optimization, which is particularly relevant for the optimal placement of beads or stampings in a structural-acoustic system.

Browder Riesz–Schauder theory for polynomially finite rank linear relations

Browder Riesz–Schauder theory for polynomially finite rank linear relations

Rough Solutions of the Einstein Constraints on Closed Manifolds without Near-CMC Conditions

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions for the Hamiltonian constraint, and topological fixed-point arguments. An important new feature of these results is the absense of the near-CMC assumption when the rescaled background metric is in the positive Yamabe class, if the freely specifiable part of the data given by the matter fields (if present) and the traceless-transverse part of the rescaled extrinsic curvature are taken to be sufficiently small. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, giving what are apparently the first non-CMC existence results without the near-CMC assumption. Standard bootstrapping arguments to increase the regularity of the conformal factor are blocked by the use of a weak background metric. In the CMC case, we recover Maxwell's rough solution results as a special case. Our results extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

An inverse problem in advection-diffusion

An inverse problem to determine an unknown velocity in two-dimensional, time-independent advection-diffusion equation from data observed at a depth-level is discussed, motivated by an ocean circulation inverse problem. A procedure by which the velocity is reconstructed from the observed data is established and, as a consequence, the uniqueness of the velocity realizing the prescribed data is proved. The procedure involves a modification of the Marchenko method in the inverse scattering theory, based on the Wiener-Hopf factorization and the Riesz-Schauder theory.

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.

Superdiagonal forms for related linear operators

The concept of superdiagonal forms for n × nmatrices T with complex entries has been extended by J. R. Ringrose [4] to the setting of compact linear operators T:X→X acting on a complex Banach space X. In a recent paper D. Koros [2] generalized Ringrose's approach to the case of compact linear operators T:X→X on a complex locally convex space X. The reason why both authors confine their attention to the class of compact linear operators is that the existence of proper closed invariant subspaces is, aside from Riesz-Schauder theory, the main tool in their construction. In the present paper it is shown that the existence of superdiagonal forms possesses a certain permanence property in the following sense.

Superdiagonal forms for completely continuous linear operators

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

On an Elliptic Boundary Value Problem with Mixed Boundary Conditions, Arising in Suspended Sediment Transport Theory

In this paper we prove the existence of an equilibrium distribution in the three-dimensional diffusion model for suspended sediment transport. Mathematically, this means that we solve an elliptic boundary value problem with mixed boundary conditions in a domain with corners. We use Riesz–Schauder theory and some of the classical Agmon–Douglis–Nirenberg estimates, among other things.

Compact operators and Banach algebras

1. Introduction. Let X be a complex Banach space and let (X) denote the Banach algebra of all bounded linear operators on X. In this paper we study subalgebras of (X) that contain non-zero compact operators and that are Banach algebras with respect to some norm dominating the operator norm. Our main aim is to give conditions for the existence of minimal one-sided ideals in . Barnes ((l) Theorem 2·2) has shown that if every element of a semi-simple Banach algebra has countable spectrum then the algebra has minimal one-sided ideals. If, in particular, is a uniformly closed sub-algebra of the compact operators on X, then every element of has countable spectrum by the Riesz–Schauder theory and so has minimal one-sided ideals. We extend this latter result by showing that if is a semi-simple uniformly closed subalgebra of (X) that contains some non-zero compact operator, then has minimal one-sided ideals. The proof depends heavily on the fact (see e.g. Bonsall (3)) that the spectral projection at a non-zero eigenvalue of a compact operator T belongs to the least closed subalgebra of (X) that contains T. The uniform norm is quite critical for Bonsall's result, but we are able to give a mild generalization which leads to conditions for the existence of minimal one-sided ideals in subalgebras of (X, Y,〈,〉) where (X, Y,〈,〉) are Banach spaces in normed duality.

Some Mathematical Aspects of Multiple Poles in the Off-Shell Scattering Amplitude

An attempt at a general theory of the Bethe-Salpeter equation in a previous work is continued, with our attention focussed on the problem of multiple poles in the off-shell scat­ tering amplitude. Basic assumptions made in the previous work reduce the problem to that of multiple poles in the resolvent of a compact operator, which is hermitian analytic as an operator-valued function of P4• Properties of compact operators are discussed on the basis of the Riesz-Schauder theory. Multiple poles are characteristic of non-normal operators, which are shown to be considerably different from hermitian operators in many points of physical importance. Several quantities useful for the understanding of multiple poles are introduced and their properties are investigated. Multiple poles in the resolvent of an ana­ lytic operator are studied as a limit of degeneracy of several simple poles.

Precompact linear operators in locally convex spaces

This paper is concerned with the properties of precompact and compact linear operators from a locally convex Hausdorff space into itself, the field of scalars being the complex number field.The Riesz–Schauder theory for the equationswhere T is a compact linear operator from the Bacach space into itself, is well known (see, for example, Banach(1), Chapter 10, §2) and has been extended to the more general setting of locally convx Hausdorff spaces by Leray (4). In particular, the ‘Fredholm alternative theorem’ remains valid.