Differentiability of the arrival time

For a monotonically advancing front, the arrival time is the time when the front reaches a given point. We show that it is twice differentiable everywhere with uniformly bounded second derivative. It is smooth away from the critical points where the equation is degenerate. We also show that the critical set has finite codimensional two Hausdorff measure. For a monotonically advancing front, the arrival time is equivalent to the level set method; a priori not even differentiable but only satisfies the equation in the viscosity sense. Using that it is twice differentiable and that we can identify the Hessian at critical points, we show that it satisfies the equation in the classical sense. The arrival time has a game theoretic interpretation. For the linear heat equation, there is a game theoretic interpretation that relates to Black-Scholes option pricing. From variations of the Sard and Lojasiewicz theorems, we relate differentiability to whether or not singularities all occur at only finitely many times for flows

The 2+12+1 convex hull of a finite set

We study $\mathbb{R}^2\oplus\mathbb{R}$-separately convex hulls of finite sets of points in $\mathbb{R}^3$, as introduced in \cite{KirchheimMullerSverak2003}. When $\mathbb{R}^3$ is considered as a certain subset of $3\times 2$ matrices, this notion of convexity corresponds to rank-one convex convexity $K^{rc}$. If $\mathbb{R}^3$ is identified instead with a subset of $2\times 3$ matrices, it actually agrees with the quasiconvex hull, due to a recent result \cite{HarrisKirchheimLin18}. We introduce "$2+1$ complexes", which generalize $T_n$ constructions. For a finite set $K$, a "$2+1$ $K$-complex" is a $2+1$ complex whose extremal points belong to $K$. The "$2+1$-complex convex hull of $K$", $K^{cc}$, is the union of all $2+1$ $K$-complexes. We prove that $K^{cc}$ is contained in the $2+1$ convex hull $K^{rc}$. We also consider outer approximations to $2+1$ convexity based in the locality theorem \cite[4.7]{Kirchheim2003}. Starting with a crude outer approximation we iteratively chop off "$D$-prisms". For the examples in \cite{KirchheimMullerSverak2003}, and many others, this procedure reaches a "$2+1$ $K$-complex" in a finite number of steps, and thus computes the $2+1$ convex hull. We show examples of finite sets for which this procedure does not reach the $2+1$ convex hull in finite time, but we show that a sequence of outer approximations built with $D$-prisms converges to a $2+1$ $K$-complex. We conclude that $K^{rc}$ is always a "$2+1$ $K$-complex", which has interesting consequences

Non-Lipschitz points and the SBV regularity of the minimum time function

This paper is devoted to the study of the Hausdorff dimension of the singular set of the minimum time function $T$ under controllability conditions which do not imply the Lipschitz continuity of $T$. We consider first the case of normal linear control systems with constant coefficients in $\mathbb{R}^N$. We characterize points around which $T$ is not Lipschitz as those which can be reached from the origin by an optimal trajectory (of the reversed dynamics) with vanishing minimized Hamiltonian. Linearity permits an explicit representation of such set, that we call $\mathcal{S}$. Furthermore, we show that $\mathcal{S}$ is $\mathcal{H}^{N-1}$-rectifiable with positive $\mathcal{H}^{N-1}$-measure. Second, we consider a class of control-affine \textit{planar} nonlinear systems satisfying a second order controllability condition: we characterize the set $\mathcal{S}$ in a neighborhood of the origin in a similar way and prove the $\mathcal{H}^1$-rectifiability of $\mathcal{S}$ and that $\mathcal{H}^1(\mathcal{S})>0$. In both cases, $T$ is known to have epigraph with positive reach, hence to be a locally $BV$ function (see \cite{CMW,GK}). Since the Cantor part of $DT$ must be concentrated in $\mathcal{S}$, our analysis yields that $T$ is $SBV$, i.e., the Cantor part of $DT$ vanishes. Our results imply also that $T$ is locally of class $\mathcal{C}^{1,1}$ outside a $\mathcal{H}^{N-1}$-rectifiable set. With small changes, our results are valid also in the case of multiple control input.Comment: 23 page

Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

We study convex integration solutions in the context of the modelling of shape-memory alloys. The purpose of the article is two-fold, treating both rigidity and flexibility properties: Firstly, we relate the maximal regularity of convex integration solutions to the presence of lower bounds in variational models with surface energy. Hence, variational models with surface energy could be viewed as a selection mechanism allowing for or excluding convex integration solutions. Secondly, we present the first numerical implementations of convex integration schemes for the model problem of the geometrically linearised two-dimensional hexagonal-to-rhombic phase transformation. We discuss and compare the two algorithms from [RZZ16] and [RZZ17].Comment: 35 pages, 14 figure

Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661