In this lecture, I want to discuss how Morse inequalities apply in more general situations where mappings are not bounded below, where there are degenerate critical points, and where we have some informa·tion of hmv solutions of the corresponding differential equations join critical· points. We then discuss an application to elliptic partial differential equation and some counterexamples which show that our estimates for the number of solutions of the elliptic problem are, in a sense, best possible. The proof of most of the results given depend upon the homotopy index of Conley [3]. However, nearly all the results can be understood without: knowing the homotopy index.

This is a report on some research done jointly with William P. Ziemer at the Centre for Mathematical Analysis. The research establishes interior regularity for a solution to a classical obstacle problem of general type~

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$. The study, begun in Keady[l981] and Keady and Kloeden[1984] of the boundary-value problem, for (\lambda/k, \psi) \[-\Delta \psi \in \lambda H (\psi - k) \hspace{.1in} in \hspace{.1in} \Omega \subset \mathbb{R}^2, \psi = 0 \hspace{.1in} on \hspace{.1in} \partial \Omega, \] is continued. Here $\Delta$ denotes the Laplacian, $H$ is the Heaviside step function and one of $\lambda$ or $k$ is a given positive constant. The solutions considered always have $\psi \gt 0$ in $\Omega$ and $\lambda/k \gt 0$, and have cores \[ A = {(x,y) \in \Omega | \psi(x,y) \gt k} \] In the special case $\Omega = B(O,R)$ , a disc, the explicit exact solutions are available. They satisfy \[ (\ast) \hspace{2in} (\psi_{m} - k)/k \rightarrow 0 \hspace{.1in} as \hspace{.1in} area(A) \rightarrow 0 , \] where $\psi_m$ is the maximum of $\psi$ over $\Omega$. Here (\ast) will be established for other domains. An adaptation of the maximum principles of Gidas, Ni and Nirenberg [1979] is an important step in establishing the above result.

We discuss the existence of steady flows of certain viscoelastic liquids under the influence of a small body force. An existence proof is obtained by showing the convergence of an iteration scheme.

In this paper we consider the Dirichlet problem for the minimal surface equation. We assume that $\Omega$ is a bounded open set in $\mathbb{R}^n$ with $C^2$ boundary $\partial \Omega$ and that $\phi$ is a continuous function on $\partial \Omega$ . Then we consider the problem : \[ (P) \hspace{.2in} Find \hspace{.1in} u \in C^2 (\Omega) \bigcap C^0 (\Omega) such \hspace{.1in} that\\ (i) u = \phi \hspace{.1in} on \hspace{.1in} \partial \Omega ,\\ (ii) u \hspace{.1in} satisfies \hspace{.1in} the \hspace{.1in} minimal \hspace{.1in} surface \hspace{.1in} equation \hspace{.1in} in \hspace{.1in} \Omega , \hspace{.1in} that \hspace{.1in} is ,\\ \sum_{i=1}^{n} D_i \left[ \frac{D_{i}u}{1 + |Du|^2} \right] = 0 \hspace{.1in} in \hspace{.1in} \Omega . \] We shall consider two aspects of this problem : firstly, whether or not solutions exist and, secondly, the regularity of solutions.