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By Michael Spivak
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Extra info for A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition
This is the object of the following result. 35. Let u be a smooth solution of γ 2 1−3γ in (−1, 1). ut = (uxx ) (1 + ux ) 2 u(x, 0) = u0 (x) = 1 − |x|, u(−1, t) = u(1, t) = 0 for t ≥ 0. Then, 1. if γ < 1, u becomes identically 0 in ﬁnite time. 2. If γ > 1, there exist two constants a and b such that 1 1 at− γ−1 ≤ u(x, t) ≤ bt− γ−1 . In particular, u is positive in (−1, 1) at any time t > 0. 3. If γ = 1, then u(x, t) goes exponentially to 0. Proof. We ﬁrst derive an equation satisﬁed by v = ux .
7. Let C be a closed Jordan curve such that there exists a parameterization for which C is of class C 1 . We then denote by C (p) the derivative of C at the point with parameter p in [a, b]. Let a ≤ t1 ≤ t2 ≤ b. Then, the length of C comprised between C(t1 ) and C(t2 ) is t2 L(C, t1 , t2 ) = |C (p)| dp. 8. There are two remarkable facts in this result. 2) above does not depend on the parameterization. 2). 9. A locally rectiﬁable Jordan curve C : [a, b] → R2 has a Euclidean parameterization if for all t ∈ [a, b], the derivative dL dt (C, a, t) exists and is identically equal to 1.
Nevertheless, we have the following theorem. 4 (Alexandrov). Let C be a closed Jordan curve. Then its range severs the plane into exactly two connected components. One is bounded and is called the interior of C denoted by Int(C) and the other one is unbounded and is the exterior of C (denoted by Ext(C)). Of course, this result is no longer valid in higher dimension and completely relies on the geometry of the plane. 2 Length of a curve We can also deﬁne the length of any part of a Jordan curve.
A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition by Michael Spivak