Supervisors info:
Γ. Μπαρμπάτης Αναπληρωτής Καθηγητής ΕΚΠΑ (Επιβλέπων), Ι. Στρατής Καθηγητής ΕΚΠΑ, Λ. Ευαγγελάτου-Δάλλα Αναπληρώτρια Καθηγήτρια ΕΚΠΑ
Summary:
In this Master thesis, we give an introduction to the theory of Schwarz
symmetrization and study its applications in problems of partial differential
equations.
We first construct the Schwarz symmetrization of a real function u, defined on
a bounded domain Ω of the N-dimension Euclidean space. The Schwarz
symmetrization of u, denoted u*, is defined on the open ball centered at the
origin and having the same volume as Ω and it is constructed by the decreasing
rearrangement of u. Then we show that for 1 p the Schwarz symmetrization
preserves the Lp norms, while for the Lp norms of the gradient of u and u*
respectively, the inequality Polya-Szego asserts:
|| u* || Lp(Ω*) || u || Lp(Ω).
The crucial tool in proving this inequality, and (generally) in this Master
thesis, is the isoperimetric inequality.
Then we deal with the Faber-Krahn inequality in Euclidean spaces. In
particular, we prove the Faber-Krahn inequality, i.e. of all sets of given
volume, the ball, and the ball alone, minimizes the first eigenvalue of the
Laplace operator with Dirichlet boundary condition:
λ1 (Ω) λ1 (Ω*).
Finally, we extend the Faber-Krahn inequality in Riemannian manifolds and prove
that it is equivalent to other known functional inequalities, such as Nash
inequality.
Keywords:
Rearrangement , Schwarz Symmetrization , Faber-Krahn Inequality, Isoperimetric Inequality, Polya-Szego Inequality
