Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces
The authors obtain new and interesting results on complete abstract second order differential equations of elliptic type in the framework of Hölder spaces. The authors consider the second order abstract differential equation \[ u''(x) + 2 B u'(x) +Au(x) = f(x), \quad x \in (0, 1)\leqno(1) \] with the boundary conditions \[ u(0) = u_0 \quad u(1) = u_1\leqno(2) \] where \(X\) is a complex Banach space and \(A, B\) are two linear operators on \(X\), \(f\in C([0, 1]; X)\) and \(u_0, u_1 \in X\). When \(f\) is Hölder continuous and the operators \(A\) and \(B\) satisfy some natural assumptions, the authors give necessary and sufficient conditions of compatibility to obtain a strict solution \(u\) and also to ensure that the strict solution has the maximal regularity property.
second order abstract differential equations
scientific article; zbMATH DE number 5359503