Is there a way to relate the variational problem where one specifies $x$ initially and finally to the initial value (Cauchy) problem where one specifies both $x$ and $p$ initially? …

What is a good book on the Calculus of Variations, for a second year PhD student?

Let $\Omega$ be a measurable set having finite Lebesgue measure. Let $p\geq 1$ and $u\in L^p(\Omega)$. Is it true that the minimum value of the real function $$ c\in \mathbb{R}^n\m …

Hi; Please I have to find the Green function of a third order boundary value problem, and i don't know if this is the same method that in the case of second order BVP ? Thank yo …

Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of $$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$ The so-calle …

I noticed that a calculus of variations problem is just an integral over a differential form. Therefore, I would think it would be possible to formulate the Euler-Lagrange equation …

I conjecture the following. Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define $$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$ $E_{\math …

Find the extremals of the functional $$J(x(t)) = \int\limits_{0}^{2}\frac{[x'(t)]^2}{x^3(t)}dt$$ subject to $x(0) =1$ and $x(2) = 4$. Does the two point boundary problem has a un …

Let $A\subset \mathbb{R}^n$ be a (measurable) bounded set, and consider the following optimization problem: minimize $P(X)$, the perimeter of a set $X$, where $X$ ranges over all C …

I seek the minimum of a certain functional which is always strictly greater than zero. The Euler-Lagrange equation is a "zeroth-order differential equation", that is, an implicit e …

A classical nice result of Euclidean geometry states that the triangles maximizing the perimeter among all inscribed triangles of a given ellipse constitue a one-parameter family. …

Start with a continuous map $f:S^{n+k} \rightarrow S^n$ (round unit spheres). The graph of $f$ lives in $S^{n+k}\times S^n$ and suppose it has a surface area (as a subspace of co- …

I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx $$ When is it possible to say that ex …

I have the following minimum problem: $$\tag{1} \min_{u\in W^{1,1}(0,B)} \int_0^B (\sqrt{1+|u^\prime (t)|^2} -a)\ u^{k-1} (t) t^{h-1}\ \text{d} t $$ (where $B>0$, $0 < a < …

I have been working on a formulation of the calculus of variations on Riemannian manifolds, formulated globally using [pullback] vector bundles and tensor bundles and their induced …