Functional equations and hypoellipticity

Degenerate Stochastic Differential Equations, Flows and Hypoellipticity

Hypoellipticity and Vanishing Theorems

Let −iLT (essentially Lie derivative with respect to T , a smooth nowhere zero real vector field) and P be commuting differential operators, respectively of orders 1 and m ≥ 1, the latter formally normal, both acting on sections of a vector bundle over a closed manifold. It is shown that if P + (−iLT ) m is elliptic then the restriction of −iLT to D ⊂ kerP ⊂ L (D is carefully specified) yields ...

The Gevrey hypoellipticity for a class of kinetic equations

In this paper, we study the Gevrey regularity of weak solutions for a class of linear and semi-linear kinetic equations, which are the linear model of spatially inhomogeneous Boltzmann equations without an angular cutoff.

Quadratic $alpha$-functional equations

In this paper, we solve the quadratic $alpha$-functional equations $2f(x) + 2f(y) = f(x + y) + alpha^{-2}f(alpha(x-y)); (0.1)$ where $alpha$ is a fixed non-Archimedean number with $alpha^{-2}neq 3$. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the quadratic $alpha$-functional equation (0.1) in non-Archimedean Banach spaces.

Nonlinear eigenvalues and analytic hypoellipticity

Motivated by the problem of analytic hypoellipticity, we show that a special family of compact non selfadjoint operators has a non zero eigenvalue. We recover old results obtained by ordinary differential equations techniques and show how it can be applied to the higher dimensional case. This gives in particular a new class of hypoelliptic, but not analytic hypoelliptic operators.