## Thesis etd-05052017-163148 |

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Thesis type

Elaborati finali per laurea triennale

Author

ROVELLINI, GIULIO

URN

etd-05052017-163148

Thesis title

"Symmetry Break" in a Minimum Problem related to Wirtinger's generalized Inequality

Department

SCIENZE MATEMATICHE, FISICHE E NATURALI

Course of study

MATEMATICA

Supervisors

**relatore**Prof. Gobbino, Massimo

Keywords

- functional inequalities
- Poincaré and Wirtinger inequalities for functions
- optimal constant in Wirtinger inequality
- Euler equation in calculus of variations

Graduation session start date

15/07/2016

Availability

Full

Summary

Once two parameters p, q > 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:

$ c \|f\|_{L^q} \le \|f'\|_{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 $.

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'\|_{L^p} / \|u\|_{L^q} $). Following an article by Dacorogna, Gangbo and Subía, we closely investigate the question of u'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.

$ c \|f\|_{L^q} \le \|f'\|_{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 $.

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'\|_{L^p} / \|u\|_{L^q} $). Following an article by Dacorogna, Gangbo and Subía, we closely investigate the question of u'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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