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| Author(s): |
Sergio Tadao Martins
Total Authors: 1
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| Document type: | Master's Dissertation |
| Press: | São Paulo. |
| Institution: | Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) |
| Defense date: | 2007-11-21 |
| Examining board members: |
Paolo Piccione;
Renato Hyuda de Luna Pedrosa;
Antonio Luiz Pereira
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| Advisor: | Paolo Piccione |
| Abstract | |
We will consider the elliptic problem $-\\Delta u + \\alpha\\chi_Du = \\lambda u in $\\Omega$, where $\\Omega$ is a domain in R^n with regular boundary, and $D \\subset\\Omega$ is a closed subset with prescribed Lebesgue measure. The motivation for this problem comes from Mechanics, where this equation models the vibrations of a composite membrane. Let $\\lambda_1(D)$ be the first eigenvalue of the problem, which is seen as a function of the set D. In this work, we will show that $\\lambda_1$ is a simple eigenvalue, and we will study the problem of minimizing $\\lambda_1(D)$ when D varies in the family of all closed subsets of $\\Omega$ with a given Lebesgue measure. More precisely, we will determine formulas for the first and the second variation of $\\lambda_1$. (AU) | |