# Metoda nivo skupa u optimizaciji oblika

Kunštek, Petar (2015) Metoda nivo skupa u optimizaciji oblika. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

This thesis studies the optimization problem in which the objective is to find the bipartition of domain that minimizes a given integral functional. The functional explicitly depends on the solution of a partial differential equation defined on a bipartion's element denoted with $\Omega$. Let $C = C(\Omega)$ be mentioned functional. In the first chapter basic properties of Banach space $W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ are introduced. Essentially, $W^{1,\infty}(\mathbb{R}^d, \mathbb{R}^d)$ can be identified as a space of a bounded Lipschitzcontinuous functions. That space is used to explain small changes of the set $\Omega$, while preserving important properties like openness and regularity of the border. Change of a set $\Omega$ is the set $\Omega' = (Id + \theta)\Omega$, where Id is an identity on $\mathbb{R}^d$ and $\theta \in W^{k,\infty}(\mathbb{R}^d, \mathbb{R}^d)$. Fixing $\Omega$ we can introduce mapping $\theta \longmapsto C(\Omega; \theta) := C(\Omega')$. To explain how small changes on $\Omega$ affect the value of the functional one can introduce shape derivative: $C'(\Omega, \theta)= \lim_{t \to 0^+} \frac{C(\Omega ; t\theta)-C(\Omega ; 0)}{t}$. The second chapter deals with technical results on a differentiability which are used in the next chapter. In the third chapter, the term local differentiability is introduced. It is used to better explain how solution can be differentiated with respect to small changes of a set $\Omega$. Within this chapter, a sufficient conditions for existence of the shape derivative $C'(\Omega, \theta)$ are given. All theory is applied in the last chapter, on a model of electric capacitor to prove existence of the local derivative.