Results about existence/uniqueness of solution to Euler-Lagrange equations?

2011-12-29T12:33:43Z

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:

What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange equations? Is there any general result?

I tried to google it, but found nothing.

Answer by Liviu Nicolaescu for Results about existence/uniqueness of solution to Euler-Lagrange equations?

2012-01-02T19:04:47Z

The so called direct method of the calculus of variations provides one such existence and uniqueness result.

Here is the gist of it. Suppose that $X$ is a reflexive Banach space, e.g. a Hilbert space or a space of the form $L^p(\Omega)$, $p\in (1,\infty)$, $\Omega$ open subset of some Euclidean space. We are given a functional $J$ on $X$, i.e., a function

$$ J : X\to (-\infty, \infty]$$

and we seek minimizers of such functionals, i.e., points $x_0\in X$ such that

$$J(x_0)=\inf_{x\in X} J(x)$$

The subset of $X$ where $J$ is finite is called the domain of $J$. It is typically described by various equalities and inequalities called constraints.

Existence Theorem. Suppose that $J$ satisfies the following conditions.

\begin{equation} \inf_{x\in X} J(x)>-\infty. \tag{A} \end{equation} \begin{equation} \mbox{The set}\;\;\lbrace J\leq t\rbrace:=\lbrace x\in X;\;\; J(x)\leq t\rbrace \;\; \mbox{is convex},\;\;\forall t\in \mathbb{R}. \tag{B} \end{equation} \begin{equation} \mbox{The set}\;\;\lbrace J\leq t\rbrace\;\; \mbox{is closed in the norm topology},\;\;\forall t\in \mathbb{R}. \tag{C} \end{equation} \begin{equation} \lim_{\|x\|\to\infty} J(x)=\infty. \tag{D} \end{equation}

Then $J$ admits at least one minimizer.

Remark. I should comment on the four conditions above. Condition (A) states that $J$ is bounded from below. Condition (B) states that $J$ is a convex function in the usual way. Condition (C) states that $J$ is lower semicontinuous in the norm topology. Under the convexity assumption this is equivalent to $J$ being lower semicontinuous with respect to the weak topology. If $J$ happens to be differentiable, then the differential of $J$ at any minimizer $x_0$ is zero. The ensuing equation $dJ(x_0)=0$ translates into the classical Euler-Lagrange equations. The minimizer postulated by the above theorem is unique provided that $J$ is strictly convex. For more about the direct method see Wikipedia and the reference therein.

In general, the objects satisfying the Euler-Lagrange equations are critical points of a functional $J: X\to\mathbb{R}$, i.e., points where the differential of $J$ vanishes. The critical points that are observable and detectable in the real world are stable and these correspond to (local) minimizers of $J$. Sometime, one is interested in not necessarily stable objects, i.e., critical points of $J$ that are not necessarily local minimizers. Morse theory is particularly good at detecting such points. All applications of this theory are based on the following principle.

Suppose that $J: H\to\mathbb{R}$ is a $C^2$ function on a Hilbert space $H$ satisfying some additional compactness assumption (e.g. the Palais-Smale condition). Suppose that there exist real numbers $a < b$ such that the sublevel sets

$$ \lbrace J\leq a\rbrace\;\;\mbox{and}\;\; \lbrace J\leq b\rbrace$$

are not homeomorphic. Then $J$ admits a critical point $x_0$ such that

$$ J(x_0)\in [a,b]. $$

For more detail see the booklet by Paul Rabinowitz, Minimax methods in critical point theory with applications to differential equations.

Results about existence/uniqueness of solution to Euler-Lagrange equations? - MathOverflow

Results about existence/uniqueness of solution to Euler-Lagrange equations?

Answer by Liviu Nicolaescu for Results about existence/uniqueness of solution to Euler-Lagrange equations?