Universität zu Köln

Schnieders, Inka (2021). Positivity and regularity of solutions to higher order Dirichlet problems on smooth domains. PhD thesis, Universität zu Köln.

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Abstract

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.