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Results tagged with calculus-of-variations
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user 3948
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
0
votes
Accepted
Minimizing functionals constrained in a box
Just to elaborate a bit on what Rahul and I mentioned in the comments.
Take the action functional to be $\int_0^1 (y'')^2 dt$, with prescribed boundary conditions $y(0) = a$, $y(1) = b$, $y'(0) = c$ …
- 35.7k
3
votes
Accepted
Variation in Einstein-Hilbert action
It is a notational short hand. (See, e.g. Appendix E in Wald's General Relativity).
Given a function $\psi$ and a one-parameter family of functions $\psi_{\lambda}$ with $\psi_0 = \psi$, the notatio …
- 35.7k
10
votes
Accepted
Is there a reason for different nomenclature on Calculus of Variations?
Your question actually is quite well answered around page 10 of Giaquinta and Hildebrandt's Calculus of Variations I: the Lagrangian formalism. The upshot is that the correct phrase you are looking fo …
- 35.7k
1
vote
Accepted
Finite energy solution for Allen -Cahn equation
Sketch of an argument:
For a fixed $x'$, let $f(x')$ denote the measure of the set $\{ u(x',x_9) \in (-1/2,1/2) \}$.
Note that for fixed $x'$ you have
$$ \int |\nabla u(x',x_9)|^2 d x_9 \geq \int …
- 35.7k
3
votes
Area-minimising hypersurface with unbounded area growth
When $n = 2$ the sort of examples you require does not exist. This is due to Fischer-Colbrie and Schoen, "The structure of complete stable minimal surfaces in 3-manifolds of non-negative scalar curvat …
- 35.7k
1
vote
Area of a deformation of a closed surface
The answer to the second question is still negative.
Let $M^3$ be the cylinder $\mathbb{S}^2 \times \mathbb{R}$. Then $\Sigma = \mathbb{S}^2\times \{0\}$ is totally geodesic in $M$, and so for any fun …
- 35.7k
8
votes
What was Weierstrass's counterexample to the Dirichlet Principle?
I think I vaguely remember what the counterexpample was, but not the details. So if someone can fill it in it'd be great! (I'm putting this in CW mode for that reason.)
The idea is based on knowing e …
4
votes
Accepted
Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?
In the vacuum case this is not greatly different from the Einstein-Hilbert action.
Let $(M,g)$ be a classical solution to the variational problem as you posed. Suppose $p\in M$ is such that $R(p) \n …
- 35.7k
3
votes
Generalized Lax-Milgram for Weak Formulation of 1D Linear Schrodinger
Your method is doomed to fail. For several reasons.
Suppose that $u$ solves the linear Schrodinger equation. Using Fourier methods it is easy to see that $\| u(\cdot,t)\|_{L^2_x}$ is conserved and in …
- 35.7k
3
votes
Accepted
How much "room" in inequality $\displaystyle \int_a^b \varphi' ov ~\mathrm{d}x \leq 0$
Let's upgrade Iosif's comment to an answer.
Let $\chi$ be a smooth bump function supported in $[-\epsilon,\epsilon]$. For any $\varphi\in U$ with support within $[a+\epsilon,a-\epsilon]$, then $\chi*\ …
- 35.7k