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Existenz und Eindeutigkeit für Lösungen des Neumann—Tricomi-Problems im ℝ2

Research paper by A. Müller-Rettkowski

Indexed on: 01 Jun '83Published on: 01 Jun '83Published in: Monatshefte für Mathematik



Abstract

In a bounded simply-connected domainG\( \subseteq \) ℝ2 a boundary value problem for a linear partial differential equation of second orderLu=f is studied. The operatorL is elliptic inG⋔{y>0}, parabolic forG⋔{y=0} and hyperbolic inG⋔{y<0}. The boundary value problem consists in findingu satisfyingLu=f inG, dnu=φ on the elliptic part of the boundary ofG, u=ψ on the noncharacteristic part (which is not empty) of the hyperbolic part of the boundary ofG.dnu denotes the conormal (with respect toL) derivative ofu. It is proved that the problem has a generalized solution in anL2-weight space. Uniqueness is otained in the class of quasiregular solutions. In order to get the results apriori estimates are proved; theorems from functional analysis are used.