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