Teoremi ulaganja Soboljevljevih prostora i primjene

Radošević, Ana (2014) Teoremi ulaganja Soboljevljevih prostora i primjene. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

In this thesis we proved the Sobolev embedding theorems of spaces $W^{k,q}(\Omega)$ in $L^p$ spaces (or even Hölder spaces, when $kq > n$, where $n$ is the dimension of $\mathbb{R}^n$) on bounded (smooth enough) domains. Among them, the Rellich-Kondrachov Theorem is particularly important, since it provides compactness of certain embeddings, which we have used in two examples to prove the existence of solutions of nonlinear boundary problems. The first example is the quasilinear elliptic PDE with homogeneous boundary condition \begin{align*} \begin{cases} -\Delta u + b(Du) + \mu u =0 &\text{on } \Omega\\ u = 0 & \text{on } \partial \Omega, \end{cases} \end{align*} where $\Omega$ is a bounded domain of class $C^2$. We assumed that $b : \mathbb{R}^n \to \mathbb{R}$ is Lipschitz continous and satisfies the growth condition $\left| b(p) \right| \leq C(\left| p \right| + 1)$ for some $C$ and all $p \in \mathbb{R}^n$. The existence of a solution $u \in H^2(\Omega) \cap H_0^1(\Omega)$ of this problem is shown using Schauder’s Fixed Point Theorem, whose assumption of compactness of the corresponding operator is secured by the compactness of the embedding $H^2(\Omega) \subset H^1(\Omega)$. The second example is the stationary Navier-Stokes problem, also with homogeneous boundary condition, on a bounded smooth domain $\Omega$, \begin{align*} \begin{cases} -\mu \Delta \textbf{u} + (D \textbf{u})\textbf{u} + \text{grad } p = \textbf{f} &\text{on } \Omega\\ div \: \textbf{u} = 0 &\text{on } \Omega\\ \textbf{u} = 0 &\text{on } \Gamma, \end{cases} \end{align*} where the vector function $\textbf{u} = (u_1, \dots , u_n)$ and the scalar function $p$ are required functions, while $\textbf{f}$ is given. Here the existence of a solution is shown using the Galerkin method, i.e. we have constructed a bounded sequence of approximate solutions and got the solution as the limit of its subsequence. To do this, we needed the compactness of the embedding $H_0^1(\Omega) \subset L^2(\Omega)$.