This paper is concerned with a comparison principle for viscosity solutions to Hamilton–Jacobi (HJ), –Bellman (HJB), and –Isaacs (HJI) equations for general classes of partial integro-differential operators. Our approach contributes to the literature in three ways: (1) We cast the Crandall–Ishii Lemma into a test function framework to tackle a wide class of second-order integro-differential operators in the spirit of the classical doubling of variables method. (2) We provide a unified approach to estimate the difference of Hamiltonians by adapting the probabilistic notion of couplings to an analytic setting. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to a variety of examples, in particular, to second-order differential operators and, more generally, generators of spatially inhomogeneous Lévy processes.

In this note, our aim is to show that families of smooth hypersurfaces of Rⁿ⁺¹ which are all “C¹ –close” enough to a fixed compact, embedded one, have uniformly bounded constants in some relevant inequalities for mathematical analysis, like Sobolev, Gagliardo– Nirenberg and “geometric” Calderòn–Zygmund inequalities.

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor. Rescaling space and time, the limit of the trajectory of the diffusion process homogenises over the periodic potential as well as over the internal degree of freedom. Around the homogenised limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton–Jacobi–Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.

In this survey we present the state of the art about the asymptotic behavior and stability of the modified Mullins–Sekerka flow and the surface diffusion flow of smooth sets, mainly due to E. Acerbi, N. Fusco, V. Julin and M. Morini. First we discuss in detail the properties of the nonlocal Area functional under a volume constraint, of which the two flows are the gradient flow with respect to suitable norms, in particular, we define the strict stability property for a critical set of such functional and we show that it is a necessary and sufficient condition for minimality under W^2,p–perturbations, holding in any dimension. Then, we show that, in dimensions two and three, for initial sets sufficiently “close” to a smooth strictly stable critical set E, both flows exist for all positive times and asymptotically “converge” to a translate of E.