# Tagged Questions

30 views

### Solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$(LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases}$$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
46 views

### Open problems in variational analysis/PDEs

I wasn't sure whether this question was more appropriate for StackExchange or Overflow, but in any case I would really appreciate it if any active researchers in the field responded. I'm a PhD ...
89 views

### Why is the integral of $\|\nabla f\|^2$ called the energy of $f$?

Let $\Omega$ be a region in $\mathbb{R}^2$ with $f:\Omega \to \mathbb{R}$ a smooth function. Why is the quantity, $$\tfrac{1}{2} \iint_{\Omega} \|\nabla f\|^2$$ Called the "energy" of $f$? I am ...
103 views

### The relationship between two definitions of star-shaped domain

There are two definitions of star-shaped domain. One is given in wikipedia as follows. Def1: A set $S$ in the Euclidean space $\mathbb{R}^n$ is called a star domain (or star-convex set, star-shaped ...
16 views

### Missing explanation in this paper of Masmoudi.

In this paper, on page 4, beginning in the line above 3.8, the authors begin a discussion of a given variational problem. I follow their argument until they begin the line of reasoning that begins ...
67 views

### Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
45 views

### Regularity of a Weak Solution

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t ...
21 views

### Higher Order Functional Equations

A common point of study is the theory of functional equations first encountered in Calculus and from there built up with the calculus of finite differences (And ultimately functional analysis) which ...
69 views

### Minimizing cost function (Eikonal)

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $S \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=S$ and ...
64 views

### Eikonal Equation [duplicate]

Given a cost function $F(x_{1},x_{2},x_{3})$ and a starting Point $A \in \mathbb{R}^{3}$ we define a function $T$ as $T(x,y,z)=\min_{\gamma} \int_{0}^{1} F(\gamma(t))dt$ such that $\gamma(0)=A$ and ...
26 views

### Coercivity for functional and complete orthonormal system

Consider with $\rho \in W^{1,2}([0,\pi])$ the following functional $$J(\rho)=\frac{1}{2}\int_{0}^{\pi}{\rho^2\,dx}$$ I know that in the $L^{2}([0,\pi])$ the coercivity condition is satisfied, but i'm ...
37 views

67 views

320 views

### Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
235 views

### Step in derivation of Euler-Lagrange equations of motion

From http://www.mathpages.com/home/kmath523/kmath523.htm Variations in $x,y,z$ and $X$ at constant $t$ are independent of $t$ (since each of these variables is strictly a function of $t$), so we ...
44 views

### How to show that this solution is a minimum.

I am reading this paper from Peter Hess: On multiple positive solutions of nonlinear elliptic eigenvalue problems. Comm. Partial Differential Equations 6 (1981), no. 8, 951–961. In the last page ...
106 views

### Null Lagrangians and “Local Degree”

Let $u: U\subset\mathbb R^n \rightarrow \mathbb R^n$ be a smooth function, $U$ bounded. Let $x_0$ and $r$ be such that $B_r(x_0)$ is disjoint from $\partial U$. Let $\eta$ be a smooth bump function ...
196 views

### Do continuous functions preserve weak*-convergence?

I am trying to comprehend the proof of a theorem in the calculus of variations (sketch: the functional $\int\limits_\Omega f(Du_j(x))~dx$ is weak*-sequentially semicontinuous if and only if $f$ is ...
67 views

### Pf. of weak lower semicontinuity for convex Lagrangians

This question is about the proof of Theorem 1 in Chapter 8 of Evans' PDE book (p. 468 in the 2nd edition). Let $u,u_k\in\mathrm{W}^{1,q}(U)$ for all $k\in\mathbb{N}$, $U\subset\mathbb{R}^n$ be open, ...
199 views

269 views

### Book Recommendation Needed: Gradient Descent, Euler-Lagrange

On a lecture note I read about Calculus of Variations faculty.uml.edu/cbyrne/cov.pdf the author talks about Euler-Lagrange equation, then continues to say "unfortunately, many times a closed form ...
According to my computations: The function which minimizes $$\int_\Omega \|\operatorname{curl} f\|^2\,dx$$ should satisfy $$\operatorname{curl}(\operatorname{curl}f) = 0$$ everywhere on $\Omega$, ...