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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  
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  
Type  
Language 
eng

Date Available 
20100825

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0071185

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201011

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International