A class of planar hypocomplex vector fields: solvability and boundary value problems. (English) Zbl 1428.35083

Meziani, A.

Rodino, Luigi G. (ed.) et al., Mathematical analysis and applications – plenary lectures. Papers based on the presentations at the 11th international ISAAC congress, Växjö, Sweden, August 14–18, 2017. Cham: Springer. Springer Proc. Math. Stat. 262, 83-107 (2018).

Summary: This paper deals with the solvability of planar vector fields \( L=A\partial_x+B\partial_y\), with \(A\) and \(B\) complex-valued function in a domain \(\Omega \subset \mathbb{R}^2\). We assume that \(L\) has a first integral \(Z\) that is a homeomorphism in \(\Omega \). To such a vector field, we associate a Cauchy-Pompeiu type operator and investigate the Hölder solvability of \(Lu=f\) and of a related Riemann-Hilbert problem when \(f\) is in \(L^p\) with \(p>2+\sigma \), where \(\sigma \) is a positive number associated to \(L\).
For the entire collection see [Zbl 1411.46003].

MSC:

35F15	Boundary value problems for linear first-order PDEs
35F05	Linear first-order PDEs
35Q15	Riemann-Hilbert problems in context of PDEs

Keywords:

first integral; Cauchy-Pompeiu-type operator