Tagged Questions
4
votes
0answers
62 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 ...
1
vote
2answers
60 views
Solving $C_1=4y^2+(y')^2+8y$
While working through the exercises in a book on the Calculus of Variations, I've hit a roadblock in trying to solve this differential equation: $C_1=4y^2+(y')^2+8y$
Let me back-up a bit and fill-in ...
2
votes
1answer
49 views
Use variation parameters method to solve: $4x^2y'' + y = 8x^{1/2}$
I've tried so hard and just get horrible, horrible equations.
$$y_p = u_1(x)x^{1/2} + u_2(x)x^{1/2}\log x$$
1
vote
0answers
28 views
variation of a final state due to changes in period (where the period is a parameter)
I have a simple ordinary differential equation
$\frac{dx}{dt}=f(x,t,p,T)$
$x(0) = x_0$, $x(T) = x_T$
where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks!
NOTE: I ...
1
vote
1answer
105 views
Separation of variables and substituion; first integral from the Euler-Differential Equation for the minimal surface problem
Let $P_1=(a,y_a),P_2=(b,y_b), y\in C^1 (a,b), y_a>0,y_b>0$
And the area integral: $\int^b_a y(x) \sqrt{1+y'(x)}dx$
From the Euler differential-equation we obtain:
$$y'=1/\alpha ...
1
vote
1answer
60 views
how to obtain Euler equation for smoothing spline minimization problem?
The question might be trivial, but I don't understand why this minimization problem in Sobolev space
$$
\min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
4
votes
1answer
122 views
Frechet Differentiabilty of a Functional defined on some Sobolev Space
How can I prove that the following Functional is Frechet Differentiable and that the Frechet derivative is continuous?
$$
I(u)=\int_\Omega |u|^{p+1} dx , \quad 1<p<\frac{n+2}{n-2}
$$
...
0
votes
0answers
142 views
Pre-requisites for the Calculus of Variations
I'm interested in working through the book : "Calculus of Variations" by Gelfand and Fomin. However, I lack the pre-requisites to do so (I'm familiar with linear algebra and one-variable calculus ...
3
votes
1answer
974 views
Simple simple Euler Lagrange Equation
Just starting a course on Lagrangian Mechanics and I'm just wondering what about the Euler-Lagrange equation, and more specifically what I'm meant to be trying to do ..
One of the questions from my ...
-1
votes
2answers
77 views
Am I allowed to move around an operator like this?
Can I take this product:
$$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$
And factor out one of the $L$'s to get:
$$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$
Where the operator $\frac{d}{dt}$ now ...
3
votes
1answer
75 views
how to solve differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$?
What's the solution of the differential equation $y^4 = k^2 (y^2 + y'^2\csc^2\alpha)$, where $y$ is a function of $x$ and $\alpha$ is a constant?
Polynomial solutions don't seem to work, because the ...
2
votes
0answers
59 views
$u''+\frac{4}{x+1}u'+\frac{2}{\left(x+1\right)^{2}}u=0$ variational solution
This is a concept solution scheme derived from a particular example that I have not been able to generalise sufficiently. The objective is to find a particular solution to a certain second-order ...
4
votes
1answer
374 views
Polar coordinates, line integrals, and the Beltrami Identity
Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
2
votes
0answers
96 views
Positive rotational symmetric solution for p-Laplacian
I have the the following problem and I just can't get my head around how to solve it.
Be $1<p<n$ and $q=\frac{np}{n-p}$, $u\in\mathcal{C}_{n,p}=\{f\in W^{1,p}_{loc}: ...
1
vote
1answer
70 views
Solve $I[y]=\int_{x_0}^{x_1}y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2} \mathrm dx$ parametrically
If $$I[y]=\int_{x_0}^{x_1}F(x,y,y') \mathrm dx$$
Where $$F=y^{-\frac{1}{2}}(1+(y')^2)^\frac{1}{2}$$
Then I have shown the Euler-Lagrange equation implies that
$$y(1+(y')^2)=2a$$
For some ...
2
votes
1answer
164 views
Find the extremals of $I[y]=\int_0^1(y')^2 \mathrm dt+\{y(1)\}^2$
Could anyone help me find the extremals of
$$I[y]=\int_0^1(y')^2 \mathrm dx+\{y(1)\}^2$$ subject to $y(0)=1$
Most crucially I can't work out how to find the boundary $x=1$. I'm trying to go back ...
1
vote
2answers
148 views
Weak lower semicontinuity of a functional on Hilbert space?
Let $H:=\left\{u\in L^2(R^N):\nabla u \in L^2(R^N)\right\}$ and a functional $$f(u)=\int_{R^N} |\nabla u|^2dx+\left(\int_{R^N} |\nabla u|^2dx\right)^2.$$
If $\{u_n\}\subset H$ is a sequence such that ...
13
votes
1answer
283 views
Hilbert's 19th problem: Why do we care?
Hilbert's 19th problem asks:
Are the solutions of regular problems in the calculus of variations always necessarily analytic?
This was proven to be true (through the work of Sergei Bernstein, ...
3
votes
3answers
259 views
Treacherous Euler-Lagrange equation
If I have an Euler-Lagrange equation:
$(y')^2 = 2 (1-\cos(y))$ where $y$ is a function of $x$ subjected to boundary conditions $y(x) \to 0$ as $x \to -\infty$ and $y(x) \to 2\pi$ as $x \to ...
6
votes
2answers
208 views
Symmetry of Solution to Classical 3-Dimensional Isoperimetric Problem
A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume".
At that time for me to write ...
5
votes
2answers
955 views
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
For an image denoising problem, the author has a functional $E$ defined
$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$
which he wants to minimize. $F$ is defined as
$$F = \|\nabla u \|^2 = u_x^2 + ...
