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Tesi etd-05052017-163148


Thesis type
Elaborati finali per laurea triennale
Author
ROVELLINI, GIULIO
URN
etd-05052017-163148
Title
"Symmetry Break" in a Minimum Problem related to Wirtinger's generalized Inequality
Struttura
SCIENZE MATEMATICHE, FISICHE E NATURALI
Corso di studi
MATEMATICA
Commissione
relatore Prof. Gobbino, Massimo
Parole chiave
  • functional inequalities
  • Poincaré and Wirtinger inequalities for functions
  • optimal constant in Wirtinger inequality
  • Euler equation in calculus of variations
Data inizio appello
15/07/2016;
Consultabilità
completa
Riassunto analitico
Once two parameters p, q &gt; 1 are fixed, we consider a Wirtinger-type inequality for functions of one variable with a null integral, i.e. an upper estimate for the q-norm of a function through the p-norm of its derivative:<br>$ c \|f\|_{L^q} \le \|f&#39;\|_{L^p} $ (where c is an adequate positive constant depending on p and q) for every $ f \in W^{1, p}(-1, 1) $ such that $ \int f = 0 $.<br>We characterize the best possible constant c (i.e. the greatest that makes the inequality true), and study the function u which actually realizes an equality (i.e. the u which minimizes functional $ F(u) = \|u&#39;\|_{L^p} / \|u\|_{L^q} $). Following an article by Dacorogna, Gangbo and Subía, we closely investigate the question of u&#39;s symmetry; in particular, we are able to prove (through careful manipulation of the Euler equation for functional F) that u is odd if and only if $ q \le 3p $. Explicit computations for u in limit cases $ p, q = 1, \infty $ are also carried out.
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