Nonregular Boundary Value Problem for the Cauchy-Riemann Operator

The purpose of the research is to assign a formally exact elliptic complex of length two to the Cauchy-Riemann Operator. The Neumann problem for this complex in a bounded domain with smooth boundary in 2  will be studied, helping therefore to solve a usual boundary value problem for the Cauchy-Riemann operator.


Introduction
Most first order linear differential operators of geometric origin are Dirac operators. Dirac operators on Riemannian manifolds are of fundamental importance in differential geometry. A ( ) k ×  -matrix D of first order scalar differential operators with constant coefficients in n  is said to be a Dirac type operator, if In this paper, we firstly restrict our discussion to a boundary value problem related to the Cauchy-Riemann operator, which is a Dirac type operator. A similar work has been done by [1] for the Fueter-operator, but using a cohomology-method. When studying a boundary value problem, we usually look for conditions which guarantee that the solution exists, is unique and depends continuously on the problem data. Let Du f x u in  , whose first component coincides with 1,0 u on the boundary of  , and where D is the Cauchy-Riemann operator.
It is important to point out that no attempt has been made here to develop any general theory. The Atiyah-Patodi-Singer index theorem drew mathematicians' attention to the so-called spectral boundary conditions for Dirac operators, thus highlighting an idea of Calderón (1963). For an excellent exposition of spectral elliptic boundary problems for Dirac operators, we refer to [2].
The scheme of the article can be declined in the following way: In Section 2, we show that boundary value problem related to the Cauchy-Riemann operator in the plane satisfies the Lopatinskii condition. The paragraph Section 3 is devoted to proving a necessary condition to the existence of a solution to our problem being our main result. To this end, finding a compatible complex to u u = on the boundary of  will be highlighted in Section 4. Before coming to some generalisations in Section 6, the corresponding Hodge theory to our problem will be handled in Section 5.

A Classical Problem
Suppose  is a bounded domain with smooth boundary in the complex plane  . Identifying  with 2  under the complex structure 1 2 z x ix = + , we consider the inhomogeneous system for an unknown function 1 2 u u iu = + in  , satisfying the boundary condition where for all 2 0 x > as well as an initial condition ( ) ( ) , the "hat" meaning Fourier transformation in 1 x . From this we read off the boundary symbol of our problem, namely is the space of all rapidly decreasing functions on the The Lopatinskii condition just amounts to saying that (2.4) is a bijective mapping for all There is no loss of generality in assuming that 1 0 ξ > .
The general solution of the homogeneous . This proves the injectivity of (2.4).
To show that (2.4) is surjective for 1 0 ξ > , we fix  . An easy computation shows that is a general solution to the system 2 v Av g ∂ − = for 2 0 x > with initial data ( ) . This solution is parametrised by a constant 2 c and it fails to belong for an arbitrary choice of 2 c . However, there is a unique constant 2 c for which it is the case. Indeed, the sum of the last two terms on the right-hand side of (2.7) is which is a rapidly decreasing function of 2 0 x ≥ ∈  . Since the second term is rapidly decreasing, the surjectivity follows.  The proof of Theorem 2.1 shows that the verification of the Lopatinskii condition is actually as hard as the construction of a parametrix to the boundary value problem.  ,

Existence of Solution
in the domain of  . To this end, we consider in Section 4 the elliptic complex related to our Cauchy-Riemann operator

A Compatibility Complex
Let us state our lemma.
Proof. The Laplacian 0 ∆ of (4.1) at step 0 is elliptic, for Since D is a square matrix of scalar differential operators with constant coefficients, we deduce that  Finally, the Laplacian 2 ∆ of (4.1) at step 2 is elliptic, for

Hodge Theory
The Hodge theory is a very important technical tool for solving partial differential equations, in particular for solving Neumann problems.
In this section, we define two very important spaces before considering a "weak version" of the Neumann problem for our elliptic complex (6.2), namely (1) It is a well known result that Neumann problems are solvable for certain classes of manifolds  , namely for manifolds which are strictly pseudoconvex with respect to the considered complex.
We now state the Hodge theory theorem related to our complex  , , We are now in a position to state the generalised lemma which is one of our results.
Lemma 6.1. The differential operators A and C fit together to form an elliptic complex over  Proof. The Laplacian 0 ∆ of (6.2) at step 0 is elliptic, for Since D is a square matrix of scalar differential operators with constant coefficients, we deduce that Finally, the Laplacian 2 ∆ of (6.2) at step 2 is elliptic, for 2 A A * ∆ = = −∆ by (6.1). 

Conclusion
In this paper, we proposed a method solving a nonregular boundary value problem for the Cauchy-Riemann operator in 2  . Nonregular in the sense, that only the component 1 u is given on the whole boundary of our domain. We even proposed an exolicit solution to our problem. The next work will be to build an explicit formula for the Laplacian of (6.2) allowing us to construct a fundamental solution of convolution type for the complex (6.2). It is precisely a homotopy formula for the complex (6.2).