On the smoothness of a solution of the first boundary-value problem for second-order degenerate elliptic-parabolic equations

Abstract

We consider the first boundary-value problem for a second-order degenerate elliptic-parabolic equation with, generally speaking, discontinuous coefficients. The matrix of leading coefficients satisfies the parabolic Cordes condition with respect to space variables. We prove that the generalized solution of the problem belongs to the Hölder space C 1+λ if the right-hand side f belongs to L p , p n > .

The Estimation of Solutions Nondivergent Elliptic-Parabolic Equations

“…Boundary value problems for the degenerate equation also were studied in the stationary case in [7]. Similar result for this type of equation in the case of coefficients satisfying the Cordes condition is obtained in adjiev, Gasimova [6].…”

“…Also the function ψ (x, t) and coefficient a ij (x, t) can tend to zero. Initial boundary problems for degenerate parabolic equations have been studied by many authors (see for example [3,4,5,6]). But the structure of the equation 1is different from that one considered in these papers.…”

The Solvability of Boundary Value Problem for Nonlinear Elliptic-Parabolic Equations

“…As to strong solvability of the first boundary value problem for elliptic-parabolic equations in the non-divergent form with smooth coefficients, we shall note in this connection the papers [4,5,6]. The similar result for the equations in the case when the coefficients satisfy the Cordes condition is obtained in [7].…”

“…Proof. The estimate (7) and the uniqueness of the solution follow from the coercive estimates. The existence of a solution is proved by considering the family of operators.…”