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UBC Theses and Dissertations

Regularity in second and fourth order nonlinear elliptic problems Cowan, Craig

Abstract

This thesis consists of six research papers.In ``Regularity of the extremal solution in a MEMS model with advection,'' we examine the equation given by $ -\Delta u + c(x) \cdot \nabla u = \lambda f(u) $ in $ \Omega$ with Dirichlet boundary conditions and where $ f(u) = (1-u)^{-2}$ or $ f(u) =e^u$. Our main result is that the associated extremal solution is smooth provided this is the case for the advection free case; $c(x)=0$. In ``Estimates on pull-in distances in MEMS models and other nonlinear eigenvalue problems'' we prove some results, which were observed numerically, regarding equations of the form $ -\Delta u = \lambda |x|^\alpha F(u) $ in $B$ where $B$ is the unit ball in $ \IR^N$. In addition we obtain upper and lower estimates on the extremal solutions associated with various nonlinear eigenvalue problems. In ``The critical dimension for a fourth order elliptic problem with singular nonlinearity,'' we examine the equation given by $ \Delta^2 u = \lambda (1-u)^{-2}$ in $ B$ with Dirichlet boundary conditions where $B$ is the unit ball in $ \IR^N$. Our main result is that the extremal solution $u^*$ is smooth if and only if $ N \le 8$. In ``Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains'' we examine the equation $ \Delta^2 u = \lambda f(u)$ in $ \Omega$ with Navier boundary conditions where $ \Omega$ is a general bounded domain in $ \IR^N$. We obtain various results concerning the regularity of the associated extremal solution. In ``Regularity of the extremal solutions in elliptic systems'' we examine the elliptic system given by $ -\Delta u = \lambda e^v$, \; $ -\Delta v = \gamma e^u$ in $ \Omega$ where $\lambda$ and $\gamma$ are positive constants and we obtain results concering the regularity of the extremal solutions. In ``Optimal Hardy inequalities for general elliptic operators with improvements'' we examine some very general Hardy inequalities. Optimal constants are obtained and we characterize the improvements of these general Hardy inequalities. In addition we prove various weighted versions of these inequalities with improvements and many other results.

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