# Minimization of regularized functional

by CSA   Last Updated June 03, 2017 08:20 AM

Background:

Let $x,y$ be points in $\mathbb{R}^D$. Let $C$ be functionals from $\mathbb{R}^D$ to $\mathbb{R}$, which are strictly convex and convex respectively. Moreover, assume that $F$ is differntiable everywhere and $C$ is differentiable everywhere except at $0$.

Define the minimization problem:

$$\underset{\gamma(0)=x,\gamma(1)=y,\gamma \in C^1([0,1],\mathbb{R}^D)}{\operatorname{argmin}} L(\gamma)=\int_a^b \|\dot{\gamma}\|_2^2 + C(\dot\gamma(t))\,dt.$$

I deduced that there exists a solution by convexity of the functional $L$ on the Banach space $C^1([0,1],\mathbb{R}^D)$, and it must be unqiue, since the sum of a convex and strictly convex functional.

Question:

How can I prove that $$\frac{\partial^n \gamma}{\partial t^n}(t) = 0; n\geq 2 .$$

Failsafe:

(If this is false what other requirements do I need to impose on $C$ to make it true, weaker than strict convexity)?

Tags :

How do you mean regularized function. In the quantum sense. Although, I am not sure, my self, what that means its either some form of linearity or positive semi-definite-ness of a form $\sigma$ additivity (normality)