Integral representations and boundary value problems for a class of systems of differential equations of elliptic type with a singular manifold. (Russian) Zbl 0701.35037

The authors prove an integral representation for solutions of \[ (1)\quad \frac{\partial^ nW}{\partial \bar z^ n}+\frac{A_ 1(z)}{a(z)\bar z+b(z)}\frac{\partial^{n-1}W}{\partial \bar z^{n-1}}+...+\frac{A_ n(z)}{(a(z)\bar z+b(z))^ n}W=\frac{f(z,\bar z)}{(a(z)\bar z+b(z))^ n}, \] in \(D^+\) \((D^-)\), where a, b are some polynomials in z(a(z)\(\neq 0)\), f is complex-valued, and \(E^+_ 0=\{z|\) \(q(z,\bar z)= a(z)\bar z+b(z)=0\), \(z\in D^+\}\), \(E^-_ 0=\{z|\) \(q(z,\bar z)=0\), \(z\in D^-\}\), \(E^{\gamma}_ 0=\{z|\) \(q(z,\bar z)=0\), \(z\in L\}\), describe the singular surfaces where various conditions will be posed. L is a smooth closed contour which divides the plane into two domains \(D^+\) and \(D^-\) with \(D^+\) containing 0.
The integral representation is constructed by writing (1) in the form \[ (2)\quad \prod^{n}_{j=1}[q(z,\bar z)\partial /\partial \bar z-\psi_ j(z)]W=f(z,\bar z), \] and breaking up the problem into \[ (3)\quad q(z,\bar z)=\partial W_ k/\partial \bar z-\psi_{n-k}(z)W_ k=W_{k+1}, \] with \(W_ 0\equiv W\), \(W_ n=f\), and \(k=1,2,...,n-2\). It is required that \((4)\quad Re \psi_ n(z)<Re \psi_{n-1}(z)<...<Re \psi_ 1(z)<0,\) in a neighborhood of \(E^+_ 0\) and \((5)\quad | a(z)\bar z+b(z)|^{-\psi_ 1(z)-1}f(z,\bar z)\in L_ p(D^+),\) \(p\geq 2.\) The solution of (3) can be written \[ (6)\quad S_ k^{\Phi_{n-k}}(W_{k+1})=| q(z,\bar z)|^{\psi_{n- k}(z)}[\Phi_{n-k}- \frac{1}{\pi}\iint_{D}\frac{W_{k+1}(\zeta)sgn(a{\bar \zeta}+b)}{| a{\bar \zeta}+b|^{\psi_{n-k}(\zeta)+1}(\zeta -z)}d\zeta], \] and the representation is \(W(z,\bar z)=\prod^{n-1}_{k=0}S_ k^{\Phi_{n-k}}(W_{k+1}),\) where if we know \(\partial^ kW/\partial \bar z^ k\) in \(D^+\), then there exist the analytic functions \(\Phi_ k\) needed to construct W.
The authors indicate how these results apply to the Riemann-Hilbert problem and to linear selfadjoint problems. In the latter case, they are able to use the representation to determine the number of free parameters in the general solution or to specify the number of supplementary conditions needed for the equation to be solvable.
Reviewer: R.Johnson


35C15 Integral representations of solutions to PDEs
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35Q15 Riemann-Hilbert problems in context of PDEs