Social and Political Dimensions of Identity

We study the interior regularity of solutions to the Dirichlet problem Lu = g in Omega, u = 0 in R-nOmega, for anisotropic operators of fractional type Lu(x) = integral(+infinity)(0) dp integral(Sn-1) da(w) 2u(x) - u(x + rho w) - u(x - rho w)/rho(1+2s). Here, a is any measure on Sn-1 (a prototype example for L is given by the sum of one-dimensional fractional Laplacians in fixed, given directions). When a is an element of C-infinity(Sn-1) and g is c(infinity)(Omega), solutions are known to be C-infinity inside Omega (but not up to the boundary). However, when a is a general measure, or even when a is L-infinity(s(n-1)), solutions are only known to be C-3s inside Omega. We prove here that, for general measures a, solutions are C1+3s-epsilon inside Omega for all epsilon > 0 whenever Omega is convex. When a is an element of L-infinity(Sn-1), we show that the same holds in all C-1,C-1 domains. In particular, solutions always possess a classical first derivative. The assumptions on the domain are sharp, since if the domain is not convex and the measure a is singular, we construct an explicit counterexample for which u is not C3s+epsilon for any epsilon > 0 - even if g and Omega are C-infinity

VIENA2: A Driving Anticipation Dataset

Action anticipation is critical in scenarios where one needs to react before the action is finalized. This is, for instance, the case in automated driving, where a car needs to, e.g., avoid hitting pedestrians and respect traffic lights. While solutions have been proposed to tackle subsets of the driving anticipation tasks, by making use of diverse, task-specific sensors, there is no single dataset or framework that addresses them all in a consistent manner. In this paper, we therefore introduce a new, large-scale dataset, called VIENA2, covering 5 generic driving scenarios, with a total of 25 distinct action classes. It contains more than 15K full HD, 5s long videos acquired in various driving conditions, weathers, daytimes and environments, complemented with a common and realistic set of sensor measurements. This amounts to more than 2.25M frames, each annotated with an action label, corresponding to 600 samples per action class. We discuss our data acquisition strategy and the statistics of our dataset, and benchmark state-of-the-art action anticipation techniques, including a new multi-modal LSTM architecture with an effective loss function for action anticipation in driving scenarios.Comment: Accepted in ACCV 201

Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimension

Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$ odd, we study the blow-up of bounded sequences $(u_k)\subset H^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation $$(-\Delta)^\frac n2 u_k=\lambda_k u_ke^{\frac n2 u_k^2}\quad \text{in }\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\frac n2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\in L^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with $(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works of Druet, Robert-Struwe and the second author, we show that if the sequence $(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence $\eta_k$ converges to the function $\eta_0(x)=\log\left(\frac{2}{1+|x|^2}\right)$, which solves the prescribed non-local $Q$-curvature equation $$(-\Delta)^\frac n2 \eta =(n-1)!e^{n\eta}\quad \text{in }\mathbb{R}^n$$ recently studied by Da Lio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$, and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if $\Lambda:=\limsup_{k\to \infty}\|(-\Delta)^\frac n4 u_k\|_{L^2}^2\ge \Lambda_1:= (n-1)!|S^n|$

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

H^s versus C^0-weighted minimizers

We study a class of semi-linear problems involving the fractional Laplacian under subcritical or critical growth assumptions. We prove that, for the corresponding functional, local minimizers with respect to a C^0-topology weighted with a suitable power of the distance from the boundary are actually local minimizers in the natural H^s-topology.Comment: 15 page

Local regularity for fractional heat equations

We prove the maximal local regularity of weak solutions to the parabolic problem associated with the fractional Laplacian with homogeneous Dirichlet boundary conditions on an arbitrary bounded open set $\Omega\subset\mathbb{R}^N$. Proofs combine classical abstract regularity results for parabolic equations with some new local regularity results for the associated elliptic problems.Comment: arXiv admin note: substantial text overlap with arXiv:1704.0756

ff-minimal surface and manifold with positive mm-Bakry-\'{E}mery Ricci curvature

In this paper, we first prove a compactness theorem for the space of closed embedded $f$-minimal surfaces of fixed topology in a closed three-manifold with positive Bakry-\'{E}mery Ricci curvature. Then we give a Lichnerowicz type lower bound of the first eigenvalue of the $f$-Laplacian on compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature, and prove that the lower bound is achieved only if the manifold is isometric to the $n$-shpere, or the $n$-dimensional hemisphere. Finally, for compact manifold with positive $m$-Bakry-\'{E}mery Ricci curvature and $f$-mean convex boundary, we prove an upper bound for the distance function to the boundary, and the upper bound is achieved if only if the manifold is isometric to an Euclidean ball.Comment: 15 page