Carleman estimates for fractional relativistic operators
Speaker 
Luz Roncal,
Basque Center for Applied Mathematics, Spain


When 
Dec 05, 2019
from 04:00 PM to 05:00 PM 
Where  LH 006 
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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 pseudodifferential 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)