On boundary value problems for a~class of ultraparabolic equations, and their applications

S. A. Tersenov

Sbornik. Mathematics, Tome 61 (1988) no. 2, pp. 529-544

Résumé

Let $\lambda_i(t)\ge\alpha>0$, and let $L$ be a strictly elliptic operator of second order in space variables $x$, with coefficients depending only on $x=(x_1,\dots,x_m)$. Using potentials, solutions of some initial-boundary value problems for the ultraparabolic equation $\sum^n_{i=1}\lambda_i(x)\frac{\partial u}{\partial t_i}=L(u)$ are constructed. These solutions belong to special Hölder spaces $H^{P,P/2}_{x\lambda}$ depending on the vector $\lambda=(\lambda_1,\dots,\lambda_n)$. By means of these notions the first boundary value problem for the equation $\sum^n_{i=1}\lambda_i\frac{\partial u}{\partial t_i}=u_{xx}\operatorname{sgn}x$ is studied in a domain containing the hyperplane $x=0$. Necessary and sufficient conditions for the existence of a solution of this problem in the spaces $H^{P,P/2}_{x\lambda}$ are given. Bibliography: 14 titles.