# Tagged Questions

155 views

### Existence of Minimizer of $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|}$

given the following functional $h(\rho) = c \|\rho\|_{3} - \int_{\mathbb R^3} \, dx \frac{\rho(x)}{|x|}$ with $\rho>0$ , $\|\rho\|_1 = 1$ and obviously $\rho\in L^1(\mathbb R^3)$. Can I see ...
142 views

### Lieb: Stability of matter, problem with variational method

I am trying to recalculate 'Stability of Matter' Paper from Lieb. I have a problem at the last step of the proof at page 3, where Lieb calculates the minimizer. I will explain my ideas and my problem ...
108 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 ...
155 views

### Variational Problem v.s. Initial Value Problem

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? As an example, ...
137 views

### Extending the variational bicomplex to Hamiltion or Hamiltion-Jacobi formalism

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 differential forms ...
555 views

### Geometric derivation of the Einsteinâ€™s field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see previous question). The standard derivation ...
455 views

### Coordinate-free derivation of the Einstein's field equation from the Hilbert action.

It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here): $$S = \int_M R \mu_g,$$ is given by ...
229 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 ...
Suppose that $G$ is a semi-simple Lie group that acts smoothly (i.e., $C^\infty$) on a smooth, finite dimensional manifold $M$. Does it follow that the action of $G$ is linearizable near any ...