# Tagged Questions

**3**

votes

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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 ...