Some classes of complex partial differential equations of second order with Fuchs operator in the main part and boundary value problems for them

Abstract

In this thesis questions of solvability of some earlier not investigated partial differential equations of second order in the plane with Fuchs operator in the main part and boundary problems for them are investigated. The following new results are received: 1\. Construction of a variety of solutions in the Sobolev class , of some classes of partial differential equations of second order in the plane with Fuchs type operator in the main part in unbounded angular domains. 2\. Algorithm of construction of continuous solutions of partial differential equations of second order in the plane with Fuchs operator in the main part. 3\. Dirichlet, Neumann and initial problem are solved for partial differential equations of second order in the plane with Fuchs operator in the main part in unbounded angular domains. 4\. Dirichlet, Neumann and an initial problem with given growth at infinity are solved for partial differential equations of second order in the plane with Fuchs operator in the main part in unbounded angular domains. The results received in the thesis can be applied in the theory of infinitesimal bendings of surfaces of positive curvature with a flattening point and to further development of the theory of boundary value problems for partial differential equations in the plane.

Die Lösbarkeit von komplexen partiellen Differentialgleichungen zweiter Ordnung mit Fuchsschem Operator als Hauptteil in ebenen Gebieten und Randwertprobleme hierzu werden behandelt. Folgende Ergebnisse werden erzielt: Konstruktion der Lösungsmannigfaltigkeit im Sobolev Raum in Winkelräumen, Algorithmus zur Konstruktion stetiger Lösungen, Lösung von Dirichlet, Neumann und Anfangswertproblemen in unbeschränkten Winkelgebieten, Dirichlet, Neumann und Anfangswertproblem unter Wachstumsbedingungen im Unendlichen. Die Ergebnisse der Arbeit sind in der Theorie der kleinen Verbiegungen von Flächen positiver Krümmung mit Flachpunkt sowie zur Weiterentwicklung der Theorie der Randwertprobleme für komplexe partielle Differentialgleichungen in der Ebene anwendbar.