## Thesis etd-04042023-161700 |

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Thesis type

Tesi di dottorato di ricerca

Author

BRIANI MEDEIROS DOS SANTOS LIMA, LUCA

URN

etd-04042023-161700

Thesis title

Shape optimization problems with competing terms

Academic discipline

MAT/05

Course of study

MATEMATICA

Supervisors

**tutor**Buttazzo, Giuseppe

Keywords

- calculus of variations
- optimization
- partial differential equations
- shape optmization

Graduation session start date

14/04/2023

Availability

Full

Summary

In this thesis we discuss several shape optimization problems in which the cost functionals are given by the product of two competing terms. For these problems, classical strategies based on the monotonicity of the energy with respect to the set inclusion are not applicable. The definitions and preliminaries required to present our results are collected and discussed in Chapter 1. In Chapter 2 we consider the p-torsional rigidity and the p-first eigenvalue of the Dirichlet p-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. In Chapter 3 we further investigate the above optimization problems, restricting ourselves to the linear case. In Chapter 4 we discuss the behaviors of the mean-to-max ratio of the p-torsion function with respect to the geometry of the domain. In Chapter 5 we study a general version of the classical Cheeger inequality. Finally, in Chapter 6 we look at the optimization problems for the shape functional defined by Jq(Ω) = P(Ω)T^q(Ω) among open sets with prescribed measure.

File

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abstract.pdf | 80.31 Kb |

Main.pdf | 978.41 Kb |

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