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Continuity of solutions of parabolic and elliptic equations. (English) Zbl 0096.06902
Als Fortsetzung früherer Arbeiten des Verf. [Proc. Natl. Acad. Sci. USA 43, 754--758 (1957; Zbl 0078.08704)] wird u.a. gezeigt: Ist $T(x,t)$ eine Lösung der parabolischen Gleichung $\nabla (C(x,t)\nabla T)=T_t$ in $n$ räumlichen Dimensionen (wobei die Eigenwerte der Matrix $C$ beschränkt sind) und ist $\vert T\vert \leq B$ in dem zugrunde gelegten Bereich $(t\geq t_0)$, so ist $$\vert T(x_1,t_1)-T(x_2,t_2)\vert \leqq BA\left\{ \left[\frac{\vert x_1-x_2\vert }{\surd \overline{t_1-t_0}}\right]^{\alpha}+\frac{t_2-t_1}{(t_1-t_0)^{\frac{\alpha}{2(1+\alpha)}}}\right\}$$ mit $t_2\geq t_1>t_0$. $A$ und $\alpha$ hängen nur von $n$ und den Schranken der Eigenwerte ab. Entsprechende Resultate gelten für die elliptischen Gleichungen $\nabla (C(x,t)\nabla T)=0$
Reviewer: C. Heinz

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