On Involutive Systems of First-order Nonlinear Partial Differential Equations

Abstract

Abstract. We study the local and microlocal analyticity of solutions u of a system of nonlinear pdes of the form

Existence of Gevrey approximate solutions for certain systems of linear vector fields applied to involutive systems of first-order nonlinear pdes

“…We would like to point out that this result, in the analytic case, was recently proved by Berhanu [6]. The case n = 1 was proved by Barostichi and Petronilho [4].…”

“…The analytic case has been recently investigated by Berhanu [6]. In contrast to the analytic case, we do not have first integrals which is one of the key points in Berhanu [6], but we are able to replace it by the existence of Gevrey approximate solutions that we have constructed here.…”

“…The nonlinear systems considered here are generalizations of the linear case described above (see Berhanu [6]). In the linear case, when the system is locally integrable, local and micro-local regularity of solutions u has been studied extensively (see, e.g., Treves [19], Journé and Trépreau [15] and references therein).…”

“…In the sequence, motivated by Berhanu [6] and Barostichi and Petronilho [4], we study Gevrey micro-regularity of solutions of certain overdetermined system of first-order pdes of the form…”

On microlocal analyticity and smoothness of solutions of first-order nonlinear PDEs

“…Results on microlocal analyticity for brackets up to order 3 were proved in [6,7] under assumptions on brackets made just at a point. In the linear case, analyticity results were proved under repeated brackets assumptions in [10,11].…”