Coercive functions from a topological viewpoint and properties of minimizing sets of convex functions appearing in image restoration

Many tasks in image processing can be tackled by modeling an appropriate data fidelity term $\mathrm{\Phi }:{\mathbb{R}}^n\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ and then solve one of the regularized minimization problems \begin{align*} &{}(P_{1,\tau}) \qquad \mathop{\rm argmin}{x \in \mathbb R^n} \big{ \Phi(x) ;{\rm s.t.}; \Psi(x) \leq \tau \big} \ &{}(P{2,\lambda}) \qquad \mathop{\rm argmin}{x \in \mathbb R^n} { \Phi(x) + \lambda \Psi(x) }, ; \lambda > 0 \end{align*} with some function $\Psi: $\mathbb{R}$^n \rightarrow $\mathbb{R}$ \cup {+\infty}$ and a good choice of the parameter(s). Two tasks arise naturally here: \begin{align*} {}& \text{1. Study the solver sets ${\rm SOL}(P{1,\tau})$and${\rm SOL}(P_{2,\lambda})$ of the minimization problems.} \ {}& \text{2. Ensure that the minimization problems have solutions.} \end{align*} This thesis provides contributions to both tasks: Regarding the first task for a more special setting we prove that there are intervals $(0,c)$ and $(0,d)$ such that the setvalued curves \begin{align*} \tau \mapsto {}& {\rm SOL}(P_{1,\tau}), ; \tau \in (0,c) \ {} \lambda \mapsto {}& {\rm SOL}(P_{2,\lambda}), ; \lambda \in (0,d) \end{align*} are the same, besides an order reversing parameter change $g:(0,c)\to (0,d)$. Moreover we show that the solver sets are changing all the time while $\tau$ runs from $0$ to $c$ and $\lambda$ runs from $d$ to $0$. In the presence of lower semicontinuity the second task is done if we have additionally coercivity. We regard lower semicontinuity and coercivity from a topological point of view and develop a new technique for proving lower semicontinuity plus coercivity. Dropping any lower semicontinuity assumption we also prove a theorem on the coercivity of a sum of functions.