Integral representations for differentiable functions (Q2712804)

If we have special elliptic systems, a Cauchy-Riemann system in the plane, the Laplace equation in higher dimensions, a Dirac system, then one can construct fundamental solutions on the one hand and derive integral representations for the solutions on the other hand. Both tools allow us to study boundary value problems for inhomogeneous systems. The author is interested in the derivation of such representations for special operators of higher-order \(\partial_k\), that is, to find an integral operator \(J_k\) as a right inverse to \(\partial^k\).NEWLINENEWLINENEWLINEFirst, he derives representations by potentials for \(C^k\)-functions. But these representations serve to solve boundary value problems, which in general, are not well-posed. His model systems are \({\partial^{m+ n}w\over\partial z^m\partial^n_{\overline z}}= f\); \(D^kw= f\); elliptic operators in \(\mathbb{C}^n\). The results can be used to understand under which additional assumptions Dirichlet type problems are solvable.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00046].