I have a bunch of papers that claim that, from the equation for shape energy: $$ F = \frac{1}{2}k_c \int (c_1+c_2-c_0)^2 dA + \Delta p \int dV + \lambda \int dA$$ one can use "meth …

It is well known that for a given volume $V$, a sphere is the shape that minimizes the surface area. I am interested in the same problem under the constraint that the shape must li …

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? …

The variational bicomplex seams to provide a modern formulation of the variational problem in terms of modern differential geometry. In particular the bigraded complex of different …

Let $M^n$ be a smooth manifold equipped with a nondegenerate Lagrangian $L:TM\to\mathbb R$, $L=L(x,y)$, $x\in M$, $y\in T_xM$. The stationary points of the corresponding integral f …

The variational approach is to seek critical points in terms the Euler-Lagrange variational derivatives $E_a(S_0)$ of a local function $S_0.$ The zero locus does not depend on c …

If ${u_n}$ is bounded in $H$（real Hilbert space）with inner product such that $(\cdot,\cdot)$, then ${\|u_n\|^2u_n}$ is bounded also. Passing a subsequence, one has that ${\|u_n\|^2 …

By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction. This means that any two closed geodesics a …

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious. General gist of the problem I have a variational p …

Problem. To give a concrete example of a geodesically reversible Finsler metric on the two-sphere that is not just the sum of a reversible Finsler metric and an exact $1$-form. So …

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 …

Given a nowhere-zero, closed $2n$-form $\Omega$ in a manifold of dimension $2n +1$, how do we know if there exists a closed $2$-form $\omega$ such that $\Omega = \omega^n$? Remark …

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the va …

Consider a boundary given by vertices (0,a), (0,0) and (1,0) (an 'L' shaped boundary). The problem is to find the equation that passes between the endpoints (0,a) (1,0) of minimum …

I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable. Let $f$ be in this class and $E$ be the set of all points w …