# Tagged Questions

142 views

### Can a function be constructed from the direction of its gradient?

Let $\Omega$ be a bounded region in $R^n$ and $J\in (L^2(\Omega))^n$ with $|J| \leq 1$ a.e. in $\Omega$. Under what conditions the equation $Du=J|Du|$, $u|_{\partial \Omega}=f$ has a solution in a ...
164 views

### Sobolev Inequality

Let $\Omega$ be a bounded region in $R^n$ and define $W:=\{ u \in H^{1}(\Omega): u(x_0)=0 \},$ where $x_0 \in \partial \Omega$ is a fixed point. Is there a constant $C$ such that ...
146 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, ...
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### 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 ...
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### A 'conjecture' on critical elliptic pde

I conjecture the following. Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define $$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$ $E_{\mathbb{R}^3}$ is defined ...
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### Willmore minimizers for genus $\geq 2$

For an immersed closed surface $f: \Sigma \rightarrow \mathbb R^3$ the Willmore functional is defined as $$\cal W(f) = \int _{\Sigma} \frac{1}{4} |\vec H|^2 d \mu_g,$$ where $\vec H$ is the mean ...
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### Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading: What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...
499 views

### Units of time in the gradient flow equation?

From the energy functional, we can derive the Euler-Lagrange equation and its corresponding gradient flow equation. My question is, what is the physical unit for time'' in the gradient flow ...
It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...