This special issue consists of a selection of papers dealing with recent progress in complex partial differential equation. It includes original and survey papers within the areas formulated in the following call for papers that was sent out at the end of 2015.
‘Within the last two decades the theory of complex partial differential equations has undergone considerable developments in several directions. One such direction deals with the study of PDEs of arbitrary orders in planar domains. General linear equations of elliptic type with model operators as leading terms are investigated. For instance problems involving polyanalytic and polyharmonic operators are of interest. Various boundary conditions lead to the analysis of corresponding fundamental solutions. For the polyharmonic operator there is a variety of such fundamental solutions: the polyharmonic Green; the polyharmonic Neumann; and many hybrids, arising through convolution of lower order particular fundamental solutions. The study of these problems are very often motivated and related to concrete physical problems in elasticity and deformation of surfaces.
In recent years explicit Green and Neumann functions were constructed for only balls and half spaces in real Euclidean spaces.
The methods used are naturally different from the classical complex analysis methods in the two-dimensional case.
In this context, Clifford analysis might be helpful. Besides providing new insight into the general theory of polyharmonic boundary value problems, explicit solutions for such problems in certain cases would be of interest.
The study of elliptic equations with singular coefficients or the study of general linear elliptic equations of higher orders are still in their infancy and just few cases are investigated.
Another direction in which PDE has recently seen significant advances is in the understanding of involutive systems of vector fields with complex-valued coefficients. Prototypes of such systems include Cauchy–Riemann structures and more generally hypoanalytic structures. Techniques such as the FBI transform are used to analyse hypoellipticity of vector field and uniqueness of Cauchy problems for certain nonlinear first-order PDEs. Extensions of classical results of complex analysis (similarity principle, F. and M. Riesz Theorem) are finding their way to solutions of elliptic equations with degeneracies.
Computational analysis based on fast Fourier transform, recursive relations in Fourier space, quadrature procedures in combination with decomposition methods, etc. have recently started to make inroads into complex PDEs.
Original papers and survey articles related to these aspects of PDE are welcome. They will contribute to a better understanding of complex differential equations and they will help build its theory. A nonexhaustive list of possible topics include:
Boundary value problems for polyanalytic, polyharmonic, and general model operator equations in the plane
Explicit polyanalytic Schwarz kernels in planar domains
Explicit polyharmonic Green, Neumann and hybrid Green functions in certain planar domains (disc, half plane etc.)
General higher order linear elliptic equations
Higher order elliptic equations with singular coefficients
Polyharmonic equations and boundary value problems in higher dimensions
Properties of solutions of systems of integrable complex vector fields
Fast algorithm methods for solving complex partial differential equations’
Riemann–Hilbert problems are also considered for hypocomplex vector fields and for complex vector fields with singular points. Gevrey regularity for Gevrey vectors for systems of vector fields and for sums of squares of vector fields is surveyed and new results are reported on. Applications to complex analysis and an extension of the Baouendi–Treves Approximation Theorem is discussed. A survey about recent progress on entire and meromorphic solutions of some linear and nonlinear second-order partial differential equations in several complex variables is included.
This collection of manuscripts represents some but not all of the major contemporary activities in the topics aimed for in the call for papers.