Asymptotic Analysis

We consider the two‐dimensional Schrödinger operator with generically bounded magnetic field, perturbed by a scalar potential $-gV$ , $g \geq 0$ being the coupling constant. We assume that $V$ is non‐negative and decays rapidly at infinity (e.g., $V({\mathbf x})\asymp \vert{\mathbf x}\vert^{-\alpha}$ , $\alpha > 2$ , as $\vert{\mathbf x}\vert\rightarrow \infty$ ). We examine the asymptotic behaviour as $g \rightarrow \infty$ of the eigenvalues situated in the gaps of the spectrum of the unperturbed operator.

This paper is the last of a series of two, where we study the asymptotics of the displacement in a thin clamped plate as its thickness tends to $0$ . In Part I, relying on the structure at infinity of the solutions of certain model problems posed on unbounded domains, we proved that the combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multi‐scale asymptotics of the displacement and optimal estimates in energy norm. The “profiles” for the boundary layer terms are solutions of such model problems. In this paper, adapting Saint‐Venant’s principle to our framework, we prove the results which we used in Part I.

Investigating more precisely the structure of the boundary layer terms, we go further in the analysis performed in Part I: the introduction of edge layer terms along the intersections of the clamped face with the top and the bottom of the plate, respectively, allows estimates in higher order norms. These edge layer terms are constructed with the help of stable asymptotics, and are the singular parts of the boundary layer terms. As a by‐product of all these investigations, we obtain expansions and estimates for the stress tensor in various anisotropic norms, and also estimates in $L^\infty$ ‐norm for the displacement field.

A general approach to relaxation for an optimal design problem for plates is proposed. Generalized minimizers in terms of Young measures are found and an attempt to reduce the design variables needed for the description of these minimizers is pursued. We examine special representations of the convex hull of a particular curve in three space dimensions where its specific properties are exploited in a fundamental way in order to obtain different descriptions of generalized minimizers. We also find the optimal relaxation by further analyzing the features of that curve.

We study the effective behaviour of a periodic array of microscopic cracks inside a homogeneous conductor. Special emphasis is placed on a rigorous study of the case in which the corresponding effective conductivity becomes nearly singular – due to the fact that adjacent cracks nearly touch. It is heuristically shown how thin clusters of such extremely close cracks may macroscopically appear as a single crack. These results have implications for our earlier work on impedance imaging.