Symposium “Analysis on Manifolds with Singularities”, Breitenbrunn 1990

Cauchy problems for hyperbolic operators often have the property, that the singularities of the initial data propagate along the bicharacteristic strips of the operator (cf. e.g. [13]). We consider, in the linear case, the situation where the bicharacteristics hit transversally a spacelike interface, which is ‘active’ in the sense that the interface condition is given via certain Fourier integral operators. Taking the identity, we obtain classical transmission conditions. A suitable functional analytic setting is furnished by the interaction concept [3], [6], [7], which covers very general mutual influences of evolution phenomena on different domains.

Let G ⊂ ℝ ⁿ be a domain with the boundary Γ which is an (n − 1)-dimensional C ^∞-manifold.

This problem was posed as a question to the first author by I. Babuška (Maryland).

Let Ω: = (−1,1) × (−1,1) ⊂ ℝ². The first kind integral equation on Ω, gives the solution of the Dirichlet problem for the Laplace equation in ℝ³\Ω̄ with Dirichlet data f given on Ω, (“screen problem”) (see [6], [16]).

This paper is a continuation of our work [1], in which some estimates were obtained for the negative spectrum of the elliptic operator , with measurable coefficients a _αβ such that and \( \operatorname{Re} \left( {L_0 u,u} \right) \equiv \operatorname{Re} \int {\quad \sum\limits_{\left| \alpha \right| \leqslant m,\left| \beta \right| \leqslant m} {\alpha _{\alpha \beta } \left( x \right)D^\alpha u\left( x \right)D^\beta \overline {u\left( x \right)} dx} } \geqslant c_0 \left| u \right|_m^2 ,\), where c ₀ is a positive constant and

Let Γ be the boundary of a simply connected bounded polyhedron Ω, in ℝ³. The harmonic double layer potential operator on Γ is defined by where do is the surface measure on Γ and n the outward pointing normal vector to Γ.

The boundary value problem L(x, u, ε) = 0, where L is an operator, u(x, ε)- the solution we look for, x ∈ D ⊂ ℝ ⁿ , is said to be singular if the asymptotic expansion is not valid everywhere in D̄. We shall consider only problems the solutions of which are slowly varying anywhere in D̄ except some narrow subdomains if ε is small. These subdomains are called boundary layers. Outside the boundary layers the solution may be represented by the asymptotic series (1). It is called the outer expansion analogously to the problem arising in hydrodynamics.

A bimetal problem is called a boundary contact problem where the contact surface between the different materials and the boundary of the entire body have common points.

It is known, that behaviour of solutions of elliptic boundary value problems near conic points of the boundary is determined by the eigenvalues of the boundary value problems polynomially depending on a spectral parameter, in regions of the unit sphere. If the positions of these eigenvalues are known, then one can draw conclusions about singularities, continuity, or smoothness of the solutions. Therefore this information is of interest for applications.

This paper is devoted to a survey of recent work of the authors on applications of singularity theory to the study of the propagation of cracks in elastic-brittle materials. In particular we consider singular situations (crack arriving to an interface between two media, for example) appearing in composite materials. Generally speaking, we only give results and some rapid indications on the proofs. Nevertheless, new results are presented in sections 4 and 6, which are a little more explicit.

This exposition deals with the conormal asymptotics transversal to the jumps of solutions of the Cauchy problem for strongly hyperbolic equations. The monograph [3] of Courant, Hilbert contains classical statements for second order equations: If the Cauchy data have a jump the solution of the Cauchy problem also jumps in the normal direction to the characteristic manifold. Many boundary value problems lead to solutions with asymptotics (cf. e.g. Schulze [13]).

We consider first order systems of strictly hyberbolic equations in \( \mathbb{R}_ + ^1 \times \mathbb{R}_ + ^n \) with initial and boundary conditions as well as first order pseudodifferential equations obtained by reducing strictly hyperbolic equations of higher order.

Differential operators with degeneracies of various sorts have been studied by a large number of mathematicians, with results ranging from those concerning very special operators to those concerning analytic and algebraic properties of general classes of operators. Much of the analysis has focused on elliptic operators, and the main concern has been to show that many properties usually associated with elliptic operators or elliptic boundary problems have analogues in a more general context. In this note we describe how a certain class of operators with uniformly controlled degeneracies may be analyzed. These are called edge operators because they arise when nondegenerate elliptic operators are written in polar coordinates around an edge of a domain, or indeed around any distinguished submanifold; however as we shall indicate, they also arise in many other geometrically natural situations. Their analysis is undertaken by constructing an algebra of pseudodifferential operators sharing the same type of degeneracy which is large enough to contain parametrices for many of the elliptic elements of the original ring of edge differential operators. These parametrices are constructed explicitly, and this allows us to conclude a great deal of detailed information about solutions of the original equations. Operators in a somewhat restricted subclass of the class of elliptic operators have very good analytic properties on a scale of weighted Sobolev or Hölder spaces for all but a discrete set of values of the weight parameter — for many values of the weight parameter the operators are semi-Fredholm, with infinite dimensional kernel or cokernel, although occasionally it will have neither hence be Fredholm. On the other hand for certain values of the weight parameter it may have both, hence the most we can conclude is that the range is closed. Simultaneously we obtain a structure theory for the generalized inverses for these operators as well as the ‘Bergman’ projectors onto the kernel or cokernel.

Suppose L is a vector field with smooth coefficients defined in a neighborhood Ω of 0 in R ^N, let σ be its principal symbol, and assume L ≠ 0 everywhere.

The report is devoted to the discussion of different statements of the Neumann problem for self-adjoint elliptic systems in domains with edges on the boundary. The aim consists in the determination of such statements that give Fredholm operators. The problem is considered in appropriate functional spaces with weighted norms. In some cases it is natural to submit the solution to the “radiation conditions” on edges.

In this paper we formulate the results on the spectral property and scattering theory for the linearized Navier-Stokes system, discribing the process of sound propagation in the non-homogeneous halfspace \( \mathbb{R}_ + ^3 = \left\{ {x = \left( {x_1 ,x_2 ,x_3 } \right):x_3 > 0} \right\}\) Here ∇ denotes the gradient operator and ∇· denotes the divergence operator, ρ denotes the fluid density, c is the sound speed, v⃗ = (v ₁, v ₂, v ₃) is the vector of vibrating volocity, p is the acoustic pressure, f⃗ = (f ₁, f ₂, f ₃) and a are solid densities corresponding the source of force and solid velocity respectively.

This article continues the author’s papers [1]–[3]. Here the Cauchy problem, boundary value and mixed problems are studied for the strictly hyperbolic in the sense of Leray-Volevich systems in the complete scale of Sobolev type spaces depending on real parameters s and τ s characterizes the smoothness of solutions in all variables and τ characterizes the additionall smoothness of the solutions in the tangential directions. When s and τ decrease the solution becomes “more generalized”, for big enough s, τ the solution is an usual classical solution of the problem in consideration. In [1]–[3] such problems were studied in the case of a single equation. About former investigations see references in [1]–[3].

In a bounded domain G ⊂ ℝ ⁿ with infinitely smooth boundary ∂G, let us consider the boundary problem

It is well known that singularities are present in solutions of boundary value problems for the Lamé equations in conical domains. It follows from the general theory [5, 9] that the solutions consist of singular terms of the form r ^α (1n r)^q F(α, ϕ, θ) (r is the distance to the vertex of the cone, ϕ and θ are the spherical angles) and a more regular term.

The application of boundary integral equation methods to potential problems with unilateral boundary condition was developped in [5], [6] and [14].

Elliptic boundary value problems in domains with edges were considered by many authors. Beside the general theory of Rempel/Schulze [16] and Melrose [14] there are papers of Kondrat’ev [10], Maz’ja/Roßmann [13], Grisvard [7], Dauge [4], Costabel/Dauge [3] and other authors. There the edge problem is attributed to the cone theory, established by Kondrat’ev in the fundamental paper [9] and further developed by Maz’ja/Plamenevskij [11].

It is well-known that the parametrices of elliptic partial differential equations on C ^∞ manifolds (say closed compact or compact with C ^∞ boundary) can be expressed by pseudo-differential operators. This implies the elliptic regularity in terms of (for instance) the standard Sobolev spaces.

In this paper we shall introduce the Lefschetz number in reduced L ²-cohomologies for proper maps of a manifold with cylinders (or cylindrical ends), which are linear with respect to the axis coordinate near infinity. We prove the homotopy invariance of this Lefschetz number when the topology is taken in the class of the maps of the same sort. Then we calculate the Lefschetz number for the case when the bases of all the cylinders are spheres.

In this lecture we present some general remarks about classes of differential operators with variable coefficients stable under perturbations of lower (in some sense) terms for which Cauchy and the mixed problem are correctly posed. The results presented here are obtained in collaboration with S.G. Gindikin and are published in [1]–[4].