Well-Posedness of the First Order of Accuracy Difference Scheme for Elliptic-Parabolic Equations in Hölder Spaces

摘要

A first order of accuracy difference scheme for theapproximate solution of abstract nonlocal boundary value problem−d2u(t)/dt2+sign(t)Au(t)=g(t),(0≤t≤1),du(t)/dt+sign(t)Au(t)=f(t),(−1≤t≤0),u(0+)=u(0−),u′(0+)=u′(0−),and u(1)=u(−1)+μfor differential equations in a Hilbert spaceHwith a self-adjoint positive definite operatorAis considered. The well-posedness of this difference scheme in Hölder spaces without a weight is established. Moreover, as applications, coercivity estimates in Hölder normsfor the solutions of nonlocal boundary value problems for elliptic-parabolic equations are obtained.