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 \begin{equation} \frac{\partial^n \gamma}{\partial t^n}(t) = 0; n\geq 2 . \end{equation}

**Failsafe:**

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

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)

- ServerfaultXchanger
- SuperuserXchanger
- UbuntuXchanger
- WebappsXchanger
- WebmastersXchanger
- ProgrammersXchanger
- DbaXchanger
- DrupalXchanger
- WordpressXchanger
- MagentoXchanger
- JoomlaXchanger
- AndroidXchanger
- AppleXchanger
- GameXchanger
- GamingXchanger
- BlenderXchanger
- UxXchanger
- CookingXchanger
- PhotoXchanger
- StatsXchanger
- MathXchanger
- DiyXchanger
- GisXchanger
- TexXchanger
- MetaXchanger
- ElectronicsXchanger
- StackoverflowXchanger
- BitcoinXchanger
- EthereumXcanger