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![]() ![]() In particular, they might be discontinuous. Note that no regularity is assumed for the coefficients of ( 3.1). Then, \(\Sigma \) satisfies ( 1.1) for \(c=1\) andįor some positive constants \(0<\lambda _1\le \lambda _2\). Simon : planes are the only entire graphs with quasiconformal Gauss map.Īligned with the classical quasiminimal terminology, we define a quasi-CMC surface as a smooth ( \(C^^3\) for which \(K\le 1\le H\) holds. Of special importance is the following quasiconformal Bernstein theorem, by L. The problem of determining which properties of minimal surfaces remain true in the quasiconformal setting of ( 1.1) has been deeply studied, see e.g. They were classically introduced by Finn (for the case of graphs), and by Osserman, who called them quasiminimal surfaces. When \(c=0\), inequality ( 1.1) corresponds to the property that the Gauss map of the surface is quasiconformal, and this defines a well-known class of surfaces. Condition ( 1.1) has its origins in some classical problems of surface theory considered, among others, by Alexandrov, Hopf, Pogorelov, Osserman, Simon and Schoen, that we explain next. Obviously, when \(\mu =0\) we obtain the CMC condition \(H=c\), but the case \(\mu \in (0,1)\) models a much more general class of surfaces. ![]() ![]() The mean curvature of each component of a soap film is constant. Soap films are made of components that are smooth surfaces. Plateau formulated a set of empirical rules, now known as Plateau’s Laws, for the formation of soap films: 1. ³ In particular, when $L' \le \min(d_0 \frac\pi2 d_1, d_1 \frac\pi2 d_0)$ where $d_0$ and $d_1$ are the lengths $|P_0P_1|$ and $|P_1P_2|$ respectively.Where \(\mu ,c\) are constants, with \(\mu <1\), and H, K denote the mean and Gaussian curvatures of the surface. Mathematically, the problem falls within the ambit of the calculus of variations. However, those equations don’t really capture how amazing and applicable calculus of variations really is so the following will be some examples of. In the first part, we discussed the idea of a functional, what it means, and how to find its extrema using the calculus of variations. But on the other hand, if the optimal solution to the relaxed problem yields intersecting $\mathcal S_i$, then the result probably does not tell us anything about the solution to the original problem. Calculus of Variations Part 2: Lines, Bubbles, and Lagrange. ² This turns out to be the case often enough - in fact, I would guess that it is always true for convex polygons. Here is a plot comparing the numerically optimized solution in blue and the cardioid in red.Īnd here is how the optimal curves vary with $L$: This yields an area of about $4.87$ units, which is greater than that of your cardioid. The integral I(y) is an example of a functional, which (more generally) is a mapping from a set of allowable functions to the reals. In your example with $n=3$ and points $P_0=(-1,0)$, $P_1=(0,0)$ and $P_2=(1,0)$, I numerically found the solution to be $r\approx1.30889$ with arcs of central angle $\theta_0=\theta_1\approx44.9^\circ$ and $\theta_2\approx260.3^\circ$. A typical problem in the calculus of variations involve finding a particular function y(x) to maximize or minimize the integral I(y) subject to boundary conditions y(a) A and y(b) B. Problems in which the Hamiltonian is a first integral solution can be somewhat easier to solve using Hamiltons Equations than the Euler-Lagrange Equations. If this curve $\mathcal C$ is simple, then it is also the closed curve that encloses the largest area, which is what was desired. Which curve (if any) maximizes the area of its enclosed region?Įxample: $p=8$, $A=(-1,0)$, $B=(0,0)$ and $C=(1,0)$Ĭandidate: let the cardioid of parametric equation $z=\frac^n A_i$ is composed of $n$ circular arcs of equal radius $r$. Problem: Among all closed curves in the plane of fixed perimeter $p$ and crossing three distinct collinear points $A$, $B$ and $C$, Given three distinct points, there is a circle crossing these points iff they are non-collinear. Of course, there are plenty of possibilities, here is one:Ī well-known theorem in geometry of the plane : The solution is well-known to be the circle, so, a way for varying the problem is to add constraints preventing the circle as solution. Of fixed perimeter, which curve (if any) maximizes the area of itsĮnclosed region? This question can be shown to be equivalent to theįollowing problem: Among all closed curves in the plane enclosing aįixed area, which curve (if any) minimizes the perimeter? » Problem can be stated as follows: Among all closed curves in the plane « The classical isoperimetric problem dates back to antiquity. ![]()
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