Carleman estimates for fractional relativistic operators

Abstract:  Let us consider the evolution equation involving the fractional relativistic operator

\(u_t(t,x)+(-\Delta + m^2)^s u(t,x)\)=\(V(t,x) u(t,x),\)   x \(\in\mathbb{R}^N,\)    t>0,                (1)

\(u(0,x)\)=\(u_0(x),\)                                                   x \(\in \mathbb{R}^N,\)

where \(s\in (0,1)\) and \(m\ge 0.\)

Our goal is to prove monotonicity estimates that may lead to control solutions to the above equation. In particular, we will present how to derive two kind of Carleman estimates:

1. First, we will use pseudo-differential calculus in order to prove Carleman estimates with quadratic exponential weight, both for parabolic and elliptic fractional operators.

2. Secondly, based on functional and spectral calculus, we show convexity estimates with linear exponential weight for solutions to equation (1).

Joint work with Diana Stan (Universidad de Cantabria, Spain) and Luis Vega (Universidad del Pa\'is Vasco and Basque Center for Applied Mathematics, Spain)