![]() In particular, they might be discontinuous. Note that no regularity is assumed for the coefficients of ( 3.1). classical result from the calculus of variations asserts that if u is a minimiser of A(u). Then, \(\Sigma \) satisfies ( 1.1) for \(c=1\) andįor some positive constants \(0<\lambda _1\le \lambda _2\). The minimal surface problem is the problem of minimising A(u). 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. plement to the classical calculus of variations. 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. Observations of the collapse of a soap-film bridge from a connected to a disconnected state are recorded. 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. Soap films always adopt the shape which minimises their elastic energy, and therefore their area, so that they turn out to be ideal in the calculus of variations, 'where we look for a. 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 interface between the beach andthe water lies at x 0. ![]() 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. Calculus of Variations 5.1 Snell’s Law Warm-up problem: You are standing at point (x1, y1) on the beach and you want to get to point (x2, y2) in the water, a few meters oshore. Calculus of variations is simultaneously a mathematical foundation for PDEs and physics, a collection of fundamental methods for solving problems in PDEs and physics, and a way to motivate. A lot of (physics) field theory is based on 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 is also the way to solve the soap bubble problem. ![]() 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. 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 $C$, For either the soap bubble problem or the brachistochrone problem the. 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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