Gerson Petronilho

On ^{∞} and Gevrey regularity of sublaplacians

In this paper we consider zero order perturbations of a class of sublaplacians on the two-dimensional torus and give sufficient conditions for global C°° regularity to persist. In the case of analytic coefficients, we prove Gevrey regularity for a generaF class of sublaplacians when the finite type condition holds.

Global s-solvability, global s-hypoellipticity and Diophantine phenomena

Abstract In this paper we consider the problem of global Gevrey solvability for a class of sublaplacians on a toruswith coefficients in the Gevrey class G s ( T N ) . For this class of operators we show that global Gevrey solvability and global Gevrey hypoellipticity are both equivalent to the condition that the coefficients satisfy a Diophantine condition.

Global hypoellipticity, global solvability and normal form for a class of real vector fields on a torus and application

The main purpose of this paper is to present a class of real vector fields defined on a torus for which the concepts of global hypoellipticity and global smooth solvability are equivalent. Furthermore, such a vector field is globally hypoelliptic if and only if its adjoint is globally hypoelliptic, and therefore we can reduce it to its normal form. As an application, we study global C∞ solvability for certain classes of sub-Laplacians.

A Cauchy-Kovalevsky Theorem for Nonlinear and Nonlocal Equations

For a generalized Camassa-Holm equation it is shown that the solution to the Cauchy problem with analytic initial data is analytic in both variables, locally in time and globally in space. Furthemore, an estimate for the analytic lifespan is provided. To prove these results, the equation is written as a nonlocal autonomous differential equation on a scale of Banach spaces and then a version of the abstract Cauchy-Kovalevsky theorem is applied, which is derived by the power series method in these spaces. Similar abstract versions of the nonlinear Cauchy-Kovalevsky theorem have been proved by Ovsyannikov, Treves, Baouendi and Goulaouic, Nirenberg, and Nishida.

Real-valued non-analytic solutions for the generalized Korteweg-de Vries equation

In both the periodic and non-periodic cases, non-analytic in time solutions to the Cauchy problem of the gKdV equation are constructed with real-valued analytic initial data when k is not a multiple of four. In the case that k = 4l, that is, the non-linearity is of the form u 4l ∂ x u, where l is a positive integer, then non-analytic in time solutions are available only for complex-valued initial data.

Simultaneous reduction of a family of commuting real vector fields and global hypoellipticity

In this paper we consider a family of commuting real vector fields on then-dimensional torus and show that it can be transformed into a family of constant vector fields provided that there is one of them which its transposed is globally hypoelliptic. We apply this result to prove global hypoellipticity for certain classes of sublaplacians.

Local uniqueness of solutions of the characteristic Cauchy problem

Local uniqueness of solutions of the characteristic Cauchy problem is shown for operators which are perturbations of operators which already have such a uniqueness.

On the hypoellipticity of degenerate elliptic boundary value problems

Abstract We use the theory of pseudodifferential operators to prove that the solutions of certain degenerate elliptic boundary value problems in the plane are smooth up to the boundary.