Positivity and regularity of solutions to higher order Dirichlet problems on smooth domains

In this thesis we investigate whether results such as a positivity preserving property or the existence of classical solutions to nonlinear problems hold for some elliptic Dirichlet problems of order $2m$. We consider a weighted polyharmonic problem $(-\Delta)^m u-\lambda w u=f$ in a bounded domain $\Omega$ with smooth boundary and $(\frac{\partial}{\partial \nu})^ku=0$ on $\partial\Omega$ for $k\in\{0,1,\dots,m-1\}$. One of the main results is the following: One assumes that there is a function $u_0$ that can be estimated from below by $d(\cdot)^m$ and which fulfills $(-\Delta)^m u_0>0$ in classical sense. Then one finds a strictly positive weight function $w$ and an interval $I\subset \mathbb{R}$, such that for $\lambda \in I$ the following holds for the Dirichlet problem described above: $f$ positive implies that $u$ is positive. The proof is based on the construction of an appropriate weight function $w$ and a corresponding strongly positive eigenfunction for the weighted polyharmonic eigenvalue problem. Then, applying a converse of the Krein-Rutman theorem for the weighted polyharmonic Dirichlet problem, one obtains the main result concerning positivity of solutions. As a special case it is shown that one finds for all smooth domains an appropriate weight function, such that the weighted bilaplace problem is positivity preserving for $\lambda$ in some small interval. Moreover, further consequences of known estimates for the polyharmonic Green function are presented. Using these estimates and regularity results, we investigate the classical solvability of a higher order semilinear Dirichlet problem. We consider the differential equation $(-\Delta)^mu +g(\cdot,u)=f$ with zero Dirichlet boundary conditions, where $g$ fulfills a sign condition $g(x,t)t\geq 0$ for all $(x,t)\in \Omega\times\mathbb{R}$ and satisfies a growth condition

Regge Finite Elements with Applications in Solid Mechanics and Relativity

University of Minnesota Ph.D. dissertation. May 2018. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); ix, 183 pages.This thesis proposes a new family of finite elements, called generalized Regge finite elements, for discretizing symmetric matrix-valued functions and symmetric 2-tensor fields. We demonstrate its effectiveness for applications in computational geometry, mathematical physics, and solid mechanics. Generalized Regge finite elements are inspired by Tullio Regge’s pioneering work on discretizing Einstein’s theory of general relativity. We analyze why current discretization schemes based on Regge’s original ideas fail and point out new directions which combine Regge’s geometric insight with the successful framework of finite element analysis. In particular, we derive well-posed linear model problems from general relativity and propose discretizations based on generalized Regge finite elements. While the first part of the thesis generalizes Regge’s initial proposal and enlarges its scope to many other applications outside relativity, the second part of this thesis represents the initial steps towards a stable structure-preserving discretization of the Einstein’s field equation

Existence, multiplicity and behaviour of solutions of some elliptic differential equations of higher order

Magdeburg, Univ., Fak. für Mathematik, Diss., 2009von Edoardo SassoneZsfassung in dt. Sprach