Potential theory for the equation of small oscillations of a~rotating fluid

B. V. Kapitonov

Sbornik. Mathematics, Tome 37 (1980) no. 4, pp. 559-579

Résumé

With the aid of potential theory the classical solvability of initial-boundary value problems is proved for the equation $$ \frac{\partial^2}{\partial t^2}\biggl(\frac{\partial^2u}{\partial x_1^2}+\frac{\partial^2u}{\partial x_2^2}+\frac{\partial ^2u}{\partial x_3^2}\biggr)+\frac{\partial^2u}{\partial x_3^2}=0 $$ in a bounded domain of the space $\Omega$, and also in the complement of this domain. For the first boundary value problem a method of obtaining estimates of solutions in uniform norms is established, with an indication of the explicit dependence of the constants on the time exhibited. Bibliography: 6 titles.