Search Results

I find it helpful in cases such as this to discretize the integrals, so that $g$ becomes a matrix with elements $g_{ij}=g(y_i,x_j)$ and $f$ becomes a vector with elements $f_i=f(x_i)$. Then $J[f]=\sum …

Quite generally, the shape $y(x)$ of the cable and the shape $w(x)$ of the suspended deck are governed by the Melan equation, a fourth order ODE. A variational formulation is given in Variational form …

A. multi-dimensional state, one-dimensional time Multi-dimensional extensions $x\in\mathbb{R}^n$ of the one-dimensional Hamilton-Jacobi-Bellman equations have been considered in Consistency of a Simp …

If you wish to interpret the gradient flow equation as an equation of physics (rather than mathematics), then you need to introduce a friction coefficient into your problem, which tells you how rapidl …

A discussion of Tonelli's contributions and their relation to Hilbert's work can be found in this AMS bulletin. The original work was published in Italian, Fondamenti di Calcolo delle Variazioni (Bolo …

Moreton's thesis work was included in a book that explores fairness measures in modeling from a variety of perspectives: Designing Fair Curves and Surfaces: Shape Quality in Geometric Modeling and Com …

"Calculus of variations" seems an accepted umbrella term; at least, looking at the corresponding Wikipedia entry, you'll recognize that most problems in this class are of the type you are looking for: …

The follow-up paper by Helfrich you are searching for is Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to s …

Since the OP states that $\lambda$ is not fixed from the outset, and assuming that $$J_G\equiv\int_{-1}^1 \cos(\tfrac{1}{2}\pi x)G(x)\,dx\neq 0,$$ we can make $I_G$ arbitrarily large by choosing $f(x …

Q: How are two Helmholtz decompositions related? A: The scalar fields differ by a harmonic function. Starting from a first decomposition $\sigma_1,\Gamma_1$, you can construct a second one by adding t …

These lecture notes by Piotr Hajłasz might have the introductory level you are looking for: The lectures will be divided into two almost independent streams. One of them is the theory of Sobolev spac …

Here is the story behind these notes, and a redirect to On the First Variation of a Varifold, W.K. Allard (1972). a quote from: Selected Works of Frederick J. Almgren

The usual way to ensure the convergence of the steepest descent formulation of the Euler-Lagrange equations, is to introduce a friction term, see The Calculus of Variations by Jeff Calder. Instead of …