Tagged Questions

0
votes
1answer
37 views

Finding the Extremals of a Functional J.

The functional $J$ is defined on smooth functions $y \colon [a,b] \to \mathbb{R}$ satisfying $y(a) = u$, $y(b) = v$ and is given by $$J[y]=\int_a^b \sqrt{y} \sqrt{1+(y')^2}\, dx.$$ I have found ...
3
votes
2answers
83 views

Derive the solution to the Lagrangian $ \mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$ \mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
1
vote
2answers
51 views

boundedness of the Green's function

Introduction: Let $A = -\Delta$ be the Laplace-Dirichlet operator, $D(A) = H_0^1(\Omega) \cap H^2(\Omega)$ where $\Omega \subset \mathbb{R}^d$ is a bounded domain. It is known that there exists a ...
3
votes
0answers
48 views

Different functional brachystochrone

Until today I thought that $$ \int_0^b \sqrt{\frac{1+y'(x)^2}{2gy(x)}} dx$$ would be the only functional to derive the brachystochrone, but in the textbook Variational Methods in Mathematical Physics ...
1
vote
1answer
98 views

closed operator, projection

Let $A: D \subset X \to X$ be a closed linear operator. X is a Banach space. Furthermore we have $\gamma: [0,1] \to \mathbb{C}$, $\gamma$ is a $C^1$ curve and $\gamma \subset \rho(A)$, where $\rho(A)$ ...
0
votes
0answers
70 views

Generalizing Jacobi's formula to continuous operators

I am trying to get a handle on the trace of an anisoptropic pseudo-differential operator $A(x,y,u(x),u(y),\nabla u(x), \nabla u(y))$. What I really need is functional derivatives of the trace log of ...
0
votes
0answers
39 views

Functional derivatives of continuous operators

I have a continuous operator of the form $A(x,y,u(x),u(y),\nabla u(x) \nabla u(y))$. I need to make some computations involving the inverse of this operator $A^{-1}$ which satisfies the relationship ...
2
votes
0answers
72 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
1
vote
0answers
66 views

Bounds on Fourier coefficients of Euclidean distance functions

I am interested in the bounding the Fourier coefficients $a_{m,n}$ of the function $f(x,y)=\sqrt{x^2+y^2}$ defined on the interval $[-1,1]^2$. I am specifically interested in understanding the ...