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Hardy-Rellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems Moradifam, Amir

Abstract

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{$\int_{B}V(x)|\nabla u²dx \geq \int_{B} W(x)u²dx,} \end{equation*} \begin{equation*} \label{two} \hbox{$\int_{B}V(x)|\Delta u|²dx \geq \int_{B} W(x)|\nabla %%@ u|^²dx+(n-1)\int_{B}(\frac{V(x)}{|x|²}-\frac{V_r(|x|)}{|x|})|\nabla 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.

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