Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Supervisors info:
Γεράσιμος Μπαρμπάτης ,καθηγητής,τμήμα μαθηματικών, ΕΚΠΑ
Νικόλαος Αλικάκος,καθηγητής,τμήμα μαθηματικών, ΕΚΠΑ
Ιωάννης Στρατής,καθηγητής,τμήμα μαθηματικών, ΕΚΠΑ
Original Title:
Ολοκληρωσιμότητα Αρνητικών Δυνάμεων Λύσεων Του Προβλήματος Saint Venant
Translated title:
The Integrability of Negative Powers of the Solution of the Saint Venant Problem
Summary:
\textlatin{In this master thesis we initiate the study of the finiteness condition} $$\int_{\Omega} u(x)^{-\beta}\leq C(\Omega, \beta)<+\infty$$\textlatin{where} $\Omega \subseteq \mathbb{R}^{n} $\textlatin{ is an open set and} $u$ \textlatin{is the solution of the Saint Venant problem}
\[
\left\{
\begin{array}{ll}
\Delta u=-1 , & \mbox{\textlatin{in} $ \Omega$,} \\
u=0 , & \mbox{\textlatin{on} $\partial\Omega$}.
\end{array}
\right.
\]
\textlatin{The central issue which we address is that of determining the range of values of the parameter } $\beta>0$ \textlatin{for which the aforementioned condition holds under various hypotheses on the smoothness of }$\Omega$ \textlatin{and demands on the nature of the constant} $C(\Omega, \beta).$ \textlatin{Classes of domains for which our analysis applies include bounded piecewise} $C^{1}$ \textlatin{domains in } $\mathbb{R}^{n}, n \geq 2$ \textlatin{, with conical singularities (in particular polygonal domains in the plane), polyhedra in } $\mathbb{R}^3$\textlatin{, and bounded domains which are locally of class} $C^2$ \textlatin{and which have (finitely many) outwardly pointing cusps.}
Main subject category:
Science
Keywords:
Sobolev spaces, Minkowski dimension, Saint Venant problem
