Showing papers by "Michael G. Crandall published in 2010"
TL;DR: For a class of norms which includes the standard p-norms on n, the authors showed that if u has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a cone function.
Abstract: Consider a function u defined on n , except, perhaps, on a closed set of potential singularities . Suppose that u solves the eikonal equation ‖Du‖ = 1 in the pointwise sense on n \, where Du denotes the gradient of u and ‖·‖ is a norm on n with the dual norm ‖·‖∗. For a class of norms which includes the standard p-norms on n , 1 < p < ∞, we show that if has Hausdorff 1-measure zero and n ≥ 2, then u is either affine or a “cone function,” that is, a function of the form u(x) = a ± ‖x − z‖∗.
33 citations
TL;DR: In this paper, it was shown that the uniqueness/existence of absolutely minimizing functions relative to a convex Hamiltonian is uniquely determined by their boundary values under minimal assumptions on the Hamiltonian.
Abstract: We show that absolutely minimizing functions relative to a convex Hamiltonian $H:\mathbb{R}^n \to \mathbb{R}$ are uniquely determined by their boundary values under minimal assumptions on $H.$ Along the way, we extend the known equivalences between comparison with cones, convexity criteria, and absolutely minimizing properties, to this generality. These results perfect a long development in the uniqueness/existence theory of the archetypal problem of the calculus of variations in $L^\infty.$
3 citations
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TL;DR: In this article, the problem of the two-stick condition was studied in the theory of eikonal equations, and the standard norm was shown to be geometrically convex and balanced.
Abstract: Let $ l =[l_0,l_1]$ be the directed line segment from $l_0\in {\mathbb R}^n$ to $l_1\in{\mathbb R}^n.$ Suppose $\bar l=[\bar l_0,\bar l_1]$ is a second segment of equal length such that $l, \bar l$ satisfy the "two sticks condition": $\| l_1-\bar l_0\| \ge \| l_1-l_0\|, \| \bar l_1-l_0\| \ge \| \bar l_1-\bar l_0\|.$ Here $\| \cdot\| $ is a norm on ${\mathbb R}^n.$ We explore the manner in which $l_1-\bar l_1$ is then constrained when assumptions are made about "intermediate points" $l_* \in l$, $\bar l_* \in \bar l.$ Roughly speaking, our most subtle result constructs parallel planes separated by a distance comparable to $\| l_* -\bar l_*\| $ such that $l_1-\bar l_1$ must lie between these planes, provided that $\| \cdot\| $ is "geometrically convex" and "balanced", as defined herein. The standard $p$-norms are shown to be geometrically convex and balanced. Other results estimate $\| l_1-\bar l_1 \|$ in a Lipschitz or H\"older manner by $\| l_* -\bar l_* \| $. All these results have implications in the theory of eikonal equations, from which this "problem of two sticks" arose.