CONTENTSIntroductionChapter 1. Asymptotic methods of integrating partial differential equationsChapter 2. The imposition of boundary conditionsChapter 3. Eigenvalue problems in shell theoryChapter 4. Momentless shell theory and its connexion with the theory of infinitesimal deflectionsChapter 5. Asymptotic integration of differential equations in the moment theory of shellsChapter 6. The effect of fixed edge conditions on the stressed state of a shellReferences

http://iopscience.iop.org/article/10.1070/RM1960v015n05ABEH001114/pdf

Some Mathematical Problems in the Linear Theory of Thin Elastic Shells

We investigate the asymptotic properties of inertial modes confined in a spherical shell when viscosity tends to zero. We first consider the mapping made by the characteristics of the hyperbolic equation (Poincare's equation) satisfied by inviscid solutions. Characteristics are straight lines in a meridional section of the shell, and the mapping shows that, generically, these lines converge towards a periodic orbit which acts like an attractor. 
We then examine the relation between this characteristic path and eigensolutions of the inviscid problem and show that in a purely two-dimensional problem, convergence towards an attractor means that the associated velocity field is not square-integrable. We give arguments which generalize this result to three dimensions. We then consider the viscous problem and show how viscosity transforms singularities into internal shear layers which in general betray an attractor expected at the eigenfrequency of the mode. We find that there are nested layers, the thinnest and most internal layer scaling with $E^{1/3}$-scale, $E$ being the Ekman number. Using an inertial wave packet traveling around an attractor, we give a lower bound on the thickness of shear layers and show how eigenfrequencies can be computed in principle. Finally, we show that as viscosity decreases, eigenfrequencies tend towards a set of values which is not dense in $[0,2\Omega]$, contrary to the case of the full sphere ($\Omega$ is the angular velocity of the system). Hence, our geometrical approach opens the possibility of describing the eigenmodes and eigenvalues for astrophysical/geophysical Ekman numbers ($10^{-10}-10^{-20}$), which are out of reach numerically, and this for a wide class of containers.

Inertial waves in a rotating spherical shell: attractors and asymptotic spectrum

Some general remarks on improperly posed problems for partial differential equations

For partial differential equations of mixed elliptic-hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet-conormal types. Such problems are of interest for applications to transonicflow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a − b − c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. c

/pdf/on-closed-boundary-value-problems-for-equations-of-mixed-1kupjinbfk.pdf

On Closed Boundary Value Problems for Equations of Mixed Elliptic-Hyperbolic Type

We reconsider here a classical “not well posed” problem, the Dirichlet problem for the wave equation for a rectangle with sides parallel to the coordinate axes. For the same domain, we also consider the Neumann, and many mixed Dirichlet–Neumann boundary value problems, including a “general mixed problem” where, at each point which is not a corner, the boundary condition is either of Dirichlet or of Neumann type.

Dirichlet, Neumann, and mixed boundary value problems for the wave equation uxx —uyy =0 for a rectangle

This paper deals with the Dirichlet problem for the hyperbolic differential equation 

$${U_{xy}} = 0$$

(1)

i. e. with the problem of determining a solution u of (1) from given values on a closed curve C.1 Simple examples show that the Dirichlet problem for a hyperbolic equation has a completely different character from that of the corresponding problem for an elliptic equation such as the potential equation 

$${U_{xx}} + {U_{yy}} = 0$$

(2)

The Dirichlet Problem for a Hyperbolic Equation

A Representation of Stieltjes Integrals by Conditionally Convergent Series

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The Dirichlet Problem for a Hyperbolic Equation

A Representation of Stieltjes Integrals by Conditionally Convergent Series