On equations of minimax type in the theory of elliptic and parabolic equations in the plane

N. V. Krylov

Sbornik. Mathematics, Tome 10 (1970) no. 1, pp. 1-19

Résumé

The existence and uniqueness of the solution in Sobolev spaces $W_p^2$ ($W_p^{2,1}$) is proved for the first boundary value problem for elliptic (parabolic) equations of the form $$ \lambda u-\inf_{\alpha\in\mathfrak U}\sup_{\beta\in\mathfrak B(\alpha)}(L_{\alpha\beta}u+f_{\alpha\beta})=f. $$ Here $L_{\alpha\beta}u=a_{ij}^{\alpha\beta}D_{ij}u+b_i^{\alpha\beta}u_{x_i}-c^{\alpha\beta}u$ and $D_{ij}u=u_{x_ix_j}$ in the elliptic case, $D_{ij}u=u_{x_ix_j}-\delta_{ij}u_t$ in the parabolic case. The subscript $p$ takes any values close to two. Bibliography: 10 titles.