We present a full and corrected proof of the stochastic verification theorem that was first obtained by X. Y. Zhou, J. Yong and X. Li, Stochastic Verification theorems within the framework of viscosity solutions, SIAM J. Control Optim., 35 (1997), pp. 243–253.

We correct the proof of Theorem 4.1 in Gozzi, Świe֒ch and Zhou [SIAM J. Control Optim., 43 (2005), pp. 2009–2019] by imposing additional conditions on the viscosity subsolution U .

It has been conjectured, on the basis of the gauge-gravity duality, that the ratio of the shear viscosity to the entropy density should be universally bounded from below by 1/4π in units of the Planck constant divided by the Boltzmann constant. Here, we prove the bound for any ghost-free extension of Einstein gravity and the field-theory dual thereof. Our proof is based on the fact that, for su...

We prove that every bounded Lipschitz function F on a subset Y of a length space X admits a tautest extension to X, i.e., a unique Lipschitz extension u : X → R for which LipUu = Lip∂Uu for all open U ⊂ X r Y . This was previously known only for bounded domains in Rn, in which case u is infinity-harmonic, that is, a viscosity solution to ∆∞u = 0, where

Here we examine a lattice-gas model that has been extended to include collective shear moves and friction interactions between particles. A shearlike field is applied by periodically translating entire subsections of the lattice with respect to one another, as opposed to biasing the individual movements of particles. Friction is introduced by forming a network of temporary bonds between particl...

Areal surface texture analysis is becoming widespread across a diverse range of applications, from engineering to ecology. In many studies silicon based impression media are used to replicate surfaces, and the fidelity of replication defines the quality of data collected. However, while different investigators have used different impression media, the fidelity of surface replication has not bee...

Drop coalescence is central to diverse processes involving dispersions of drops in industrial, engineering, and scientific realms. During coalescence, two drops first touch and then merge as the liquid neck connecting them grows from initially microscopic scales to a size comparable to the drop diameters. The curvature of the interface is infinite at the point where the drops first make contact...

A drop impacting a solid surface with sufficient velocity will emit many small droplets creating a splash. However, splashing is completely suppressed if the surrounding gas pressure is lowered. The mechanism by which the gas affects splashing remains unknown. We use high-speed interference imaging to measure the air beneath all regions of a spreading viscous drop as well as optical absorption ...

In this paper we show how to use a Tug-of-War game to obtain existence of a viscosity solution to the infinity laplacian with nonhomogeneous mixed boundary conditions. For a Lipschitz and positive function g there exists a viscosity solution of the mixed boundary value problem,