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Note that the second integral is the integral over the triangle $\newcommand{\bR}{\mathbb{R}}$ $$ \Delta_T=\big\{ (s,t)\in\bR^2;\;\;0\leq s\leq t\leq T\big\}. $$ Fubini's theorem shows that for any i …

I think that a straight variational approach has a good chance of yielding the desired conclusion. Fix two points $z_0,z_1$ on the boundary of the unit disk. Denote by $C$ the positively oriented …

The book of Gelfand and Fomin is a good place to start. It worked for me. I would like to include another nice and short source namely Chapter 19, vol. II of Feynman's Lectures on Physics. If …

Suppose that $\newcommand{\eF}{\mathscr{F}}$ $\newcommand{\bR}{\mathbb{R}}$ $\eF: C\to \bR$ is a convex functional defined on a closed convex subset $C$ of a say real Banach space $U$. (You can all …

Under certain nondegeneracy conditions, a Lagrangian $L$ on a manifold $M$, i.e., a function on the total space $TM$ of the tangent bundle of $M$ defines a diffeomorphism $$\Psi_L: TM\to T^*M$$ k …

$J$ is not coercive in $W^{1,2}$ For that to happen you need to show that $\Vert u_n\Vert_{1,2}\to \infty$ implies $J(u_n)\to \infty$. Take for example the function $u_n$ which is identically $0$ …

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 Hilb …