Extrema of a Differentiable Functions on Manifolds

Subjects: Metric and Normed Spaces
Links: Fréchet-Derivative, Lagrange Multipliers, Implicit Function Theorem, Existence of Maximums and Minimums of Functions on Metric Spaces

Let $\mathrm{\Omega }$ be an open subset of an Banach space $Y$, $\phi :\mathrm{\Omega }\to {\mathbb{R}}^m$ class ${\mathcal{C}}^1$, $c\in {\mathbb{R}}^m$ and $M:={\phi }^{-1}[\{c\}]=\{u\in \mathrm{\Omega }\mid \phi (u)=c\}$.

Prop: If $c$ is a regular value of $\phi$ then for each $u\in M$, $$T_uM = {\sigma'(0) \mid \sigma\in \Gamma_u(M)}$$where ${\mathrm{\Gamma }}_u(M)$ is the set of all functions $\sigma :(-\epsilon ,\epsilon )\to Y$ of class ${\mathcal{C}}^1$ such that $\sigma (0)=u$ and $\sigma (t)\in M$ for each $(-\epsilon ,\epsilon )$.

Def: A subset $M$ of a Banach space $Y$ it is called a submanifold of $Y$ of class ${\mathcal{C}}^k$ and of codimension $m$ if it exists a function $\phi :\mathrm{\Omega }\to {\mathbb{R}}^m$ of class ${\mathcal{C}}^k$ defined on an open subset $\mathrm{\Omega }$ of $Y$ that contains $M$ and a regular value $c\in {\mathbb{R}}^m$ of $\phi$ such that $M=\{u\in \mathrm{\Omega }\mid \phi (u)=c\}$.

Prop: Let $M$ be submanifold of a Banach space $Y$, $\mathrm{\Omega }$ an open subset of $Y$ that contains $M$, and $g:\mathrm{\Omega }\to \mathbb{R}$ is of class ${\mathcal{C}}^1$. If $u$ is a local maximum or local minimum on $M$, then $$g'(u) v = 0 \qquad \forall v \in T_u M$$
Def: Let $M$ be a submanifold of a Banach space $Y$, $\mathrm{\Omega }$ is an open subset of $Y$ that contains $M$, and $g:\mathrm{\Omega }\to \mathbb{R}$ is a function of class ${\mathcal{C}}^1$. We say that a point $u\in M$ is a critical point of $g$ on $M$ if$$g'(u) v = 0 \qquad \forall v \in T_u M$$
We see that local maxima and local minima are critical points of $g$ on $M$.

Lagrange Multipliers

Let $\mathrm{\Omega }$ an open set of ${\mathbb{R}}^n$, $\phi =({\phi }_1,\dots ,{\phi }_m):\mathrm{\Omega }\to {\mathbb{R}}^m$ is a function of class ${\mathcal{C}}^1$, $c\in {\mathbb{R}}^m$ is a regular value of $\phi$ and $M:=\{x\in {\mathbb{R}}^n\mid \phi (x)=c\}$. Let $g:\mathrm{\Omega }\to \mathbb{R}$ a function of class ${\mathcal{C}}^1$. Then $x_0$ is a critical point of $g$ on $M$ iff there exists unique ${\lambda }_1,\dots ,{\lambda }_m\in \mathbb{R}$ such that $$\nabla g(x_0) = \sum_{i = 1}^m \lambda_i \nabla\varphi_i(\xi) \qquad \text{and} \qquad \varphi(\xi) = c$$
Which cam be done in it in a really simple way