In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of RN with prescribed measure m attains its maximum on the union of two disjoint balls of measure m/2. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

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This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J. Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation is pedestrian dynamics in a small corridor where the propagation of people in a part of the corridor can be either left or rightgoing. Under the assumptions of propagation of chaos and mean-field limit, we derive a master equation and the corresponding meanfield kinetic and macroscopic models. Steady--states are computed and analyzed analytically and exhibit the possibility of multiple meta-stable states and hysteresis.

In this note we consider regularity theory for a fractional p-Laplace operator which arises in the complex interpolation of the Sobolev spaces, the $H^{s,p}$-Laplacian. We obtain the natural analogue to the classical p-Laplacian situation, namely $C^{s+α}$-regularity for the loc homogeneous equation.