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On the regularity of the variational solution of the Tricomi problem in the elliptic region. (English) Zbl 0573.35067
Betrachtet wird das Tricomi Problem (und Neumann Problem) für \(Tu := yu_{xx} + u_{yy}\) in einem Gebiet \(G\), dessen Rand \(\partial G\) durch \(\Sigma\), \(\Gamma\) und \(\Gamma_1\) gebildet wird. \(\Sigma\) liegt in der oberen Halbebene, verbindet die Punkte \(A(0,0)\) und \(B(1,0)\), \(\Gamma\) und \(\Gamma_1\) sind Charakteristiken durch \(A\) und \(B\). Es wird gefordert, daß \(\Sigma\) für kleine \(y>0\) mit den Geraden \(x=0\) und \(x=1\) zusammenfällt und \(\Omega = G \cap \{y>0\}\) konvex ist. Es ist bekannt, daß sich die Vorgaben von \(u\) bzw. \(d_ nu\) auf \(\Gamma\) auf Forderungen an \(u\) auf der parabolischen Linie transformieren lassen. Für das Tricomi Problem erhält man z.B. ein elliptisch-parabolisches Randwertproblem \[ Tu = yu_{xx} + u_{yy} = f \text{ in } \Omega, u(t,0)/(x-t)^{-2/3}u = 0 \text{ auf } \Sigma, \tag{\(*\)} \]
\[ Lu := u_ y(x,0) - k(d/dx) \int^{x}_{0}u(t,0)(x-t)^{-2/3}\,dt = \phi(x) \text{ für} \]
\[ x\in I = \{(x,0)| \;0\leq x\leq 1\},\quad k \text{Konstante.} \] Die Existenz und Eindeutigkeit einer schwachen Lösung von (\(*\)) aus \[ H^1_ y(\Omega) = \{v| \;v\in L^ 2(\Omega), \;y^{1/2}v_ x\in L^ 2(\Omega),\;v_ y\in L^ 2(\Omega)\} \] ist gesichert. Bezeichnet \(H^ 2_ y(\Omega)=\{v| \;v\in H^ 1_ y(\Omega)\), \(y^{1/2}v_{xx}\in L^ 2(\Omega)\), \(v_{xy}\in L^ 2(\Omega)\), \(y^{-1/2}v_{yy}\in L^ 2(\Omega)\}\), \(W=\{v| v\in H^ 2_ y(\Omega)\), \(v=0\) auf \(\Sigma\}\), \(L^ 2(y^{-1/2}) = \{f| f\in D'(\Omega),\quad y^{-1/2}f\in L^ 2(\Omega)\},\quad_ 0H^{2/3}(I)=\{\psi | \psi \in H^{2/3}(I)\), \(\psi (0)=0\}\), so wird folgende Frage untersucht:
Gegeben sei \(f\in L^ 2(y^{-1/2})\). \(\phi \in_ 0H^{2/3}(I)\), ist dann die schwache Lösung \(u\in H^ 2_ y(\Omega)\) von (\(*\)) auch Element des Raumes W ? Es zeigt sich, daß im allgemeinen gilt \(u\not\in W\). Das Singularitätsverhalten der schwachen Lösung u von (\(*\)) im Eckpunkt \(B(1,0)\) wird eingehend untersucht.
Reviewer: M.Schneider
MSC:
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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