J. Solà-Morales

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay

Abstract We are concerned with hypersurfaces of ℝ N {\mathbb{R}^{N}} with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ℝ N {\mathbb{R}^{N}} with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ℝ 2 {\mathbb{R}^{2}} with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

Convergence to steady-state and boundary layer profiles in a linear chromatography system

We study the hyperbolic system of equations of the so-called linear transport model in a true moving bed chromatography device with four ports. By using methods based on a suitable energy functional we show that all solutions approach exponentially a unique steady-state solution. Then, with the use of asymptotic analysis techniques we calculate the limit profiles of these steady-state solutions when the mass transfer coefficient between the liquid and solid phases tends to infinity. Along this singular limit sharp boundary layers appear near some ports. We are able to obtain explicit and simple formulas for these limit profiles.