 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 HardyRellich inequalities and the critical dimension...
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
HardyRellich 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 HardyRellich 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 HardyRellich 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 u²dx \geq \int_{B} W(x)\nabla %%@ u^²dx+(n1)\int_{B}(\frac{V(x)}{x²}\frac{V_r(x)}{x})\nabla u²dx. \end{equation*} This characterization makes a very useful connection between Hardytype inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and HardyRellich type inequalities. In section 4, with a similar approach, we present various classes of HardyRellich 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 $\Delta² u=\frac{\lambda}{(1u)², which models a simple MicroElectromechanical System (MEMS) device on a ball B\subset R^N, under Dirichlet or Navier boundary conditions. We show that u* is regular provided N ≤ 8 while u_{\lambda^*} is singular for N ≥ 9. In section 7, by a rigorous mathematical proof, we show that the extremal solutions of the bilaplacian with exponential nonlinearity is singular in dimensions N ≥ 13. In the third part, motivated by the theory of selfduality we propose new templates for solving nonsymmetric linear systems. Our approach is efficient when dealing with certain illconditioned and highly nonsymmetric systems.
Item Metadata
Title  HardyRellich inequalities and the critical dimension of fourth order nonlinear elliptic eigenvalue problems 
Creator  Moradifam, Amir 
Publisher  University of British Columbia 
Date Issued  2010 
Description 
This thesis consists of three parts and manuscripts of seven research papers studying improved Hardy and HardyRellich inequalities, nonlinear eigenvalue problems, and simultaneous
preconditioning and symmetrization of linear systems.
In the first part that consists of three research papers we study improved Hardy and HardyRellich inequalities. In sections 2 and 3, we give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rn, n ≥ 1, so that the following inequalities hold for all u \in C_{0}^{\infty}(B):
\begin{equation*} \label{one}
\hbox{$\int_{B}V(x)\nabla u²dx \geq \int_{B} W(x)u²dx,}
\end{equation*}
\begin{equation*} \label{two}
\hbox{$\int_{B}V(x)\Delta u²dx \geq \int_{B} W(x)\nabla %%@
u^²dx+(n1)\int_{B}(\frac{V(x)}{x²}\frac{V_r(x)}{x})\nabla
u²dx.
\end{equation*}
This characterization makes a very useful connection between Hardytype inequalities and the oscillatory behavior of certain ordinary differential equations. This allows us to improve, extend, and unify many results about Hardy and HardyRellich type inequalities. In section 4, with a similar approach, we present
various classes of HardyRellich inequalities on H²\cap H¹₀
The second part of the thesis studies the regularity of the extremal solution of fourth order semilinear equations. In sections 5 and 6
we study the extremal solution u_{\lambda^*}$ of the semilinear biharmonic equation $\Delta² u=\frac{\lambda}{(1u)², which models a simple MicroElectromechanical 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 selfduality we propose new templates for solving nonsymmetric linear systems. Our approach is efficient when dealing with certain illconditioned and highly nonsymmetric systems.

Genre  Thesis/Dissertation 
Type  Text 
Language  eng 
Date Available  20100825 
Provider  Vancouver : University of British Columbia Library 
Rights  AttributionNonCommercialNoDerivatives 4.0 International 
DOI  10.14288/1.0071185 
URI  
Degree  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Graduation Date  201011 
Campus  UBCV 
Scholarly Level  Graduate 
Rights URI  
Aggregated Source Repository  DSpace 
Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International