3
votes
0answers
74 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
0
votes
0answers
114 views

Third variation of area of a minimal surface

There is a formula for the third variation of area on page 96 of Nitsche's book, Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the page it is good for normal ...
2
votes
3answers
272 views

Symmetry Properties of Minimizers - Calculus of Variations

What methods are there to show symmetry properties of the minimizer of a problem $\inf_{u\in X}\mathcal{F}(u)$ in the calculus of variations? In general, the symmetry properties of $\mathcal{F}$ do ...
4
votes
1answer
120 views

Deriving Helfrich's shape equation for closed membranes

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 "methods of variational ...
7
votes
1answer
265 views

Formulating the calculus of varations with exterior calculus

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 equations using exterior ...
4
votes
1answer
247 views

Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?

This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard ...
8
votes
1answer
524 views

Maximal tetrahedra inscribed in ellipsoid

Pietro Majer quoted the theorem of Michel Chasles in his MO question, "Convex curves with many inscribed triangles maximizing perimeter," which states that the triangles of maximum perimeter inscribed ...
10
votes
2answers
2k views

Surface equivalent of catenary curve

A catenary curve is the shape taken by an idealized hanging chain or rope under the influence of gravity. It has the equation $y= a \cosh (x/a)$. My question is: What is the shape taken by an ...