Hardy-Rellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems
Amir Moradifam · cIRcle (University of British Columbia) · 2010
This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and Hardy-Rellich inequalities, nonlinear eigenvalue problems, and simultaneous preconditioning and symmetrization of linear systems. In the first part that consists of three research papers we study improved Hardy and Hardy-Rellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n ≥ 1, so that the following inequalities hold for all u \in C_{0}^{\infty}(B): \begin{equation*} \label{one} \hbox{$\int_{B}V(x)| abla u²dx \geq \int_{B} W(x)u²dx,} \end{equation*} \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\Delta u|²dx \geq \int_{B} W(x)| abla %%@ u|^²dx+(n-1)\int_{B}(\frac{V(x)}{|x|²}-\frac{V_r(|x|)}{|x|})| abla u|²dx. \end{equation*} This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and Hardy-Rellich type inequalities. In section 4, with a similar approach, we present various classes of Hardy-Rellich inequalities on H²\cap H¹₀ The second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6 we study the extremal solution u_{\lambda^*}$ of the semilinear biharmonic equation $\Delta² u=\frac{\lambda}{(1-u)², which models a simple Micro-Electromechanical System (MEMS) device on a ball B\subset R^N, under Dirichlet or Navier boundary conditions. We show that u* is regular provided N ≤ 8 while u_{\lambda^*} is singular for N ≥ 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N ≥ 13. In the third part, motivated by the theory of self-duality we propose new templates for solving non-symmetric linear systems. Our approach is efficient when dealing with certain ill-conditioned and highly non-symmetric systems.