Free boundary value problems for abstract elliptic equations and applications

Free boundary value problem for abstract elliptic equations with variable coefficients is studied. The equations involve linear operators in Banach space $E$. The uniform maximal regularity properties and Fredholmness of this problem are obtained in $E$-valued Hölder spaces. It is proven that the corresponding differential operator is positive and is a generator of an analytic semigroup. In application, the maximal regularity properties of Cauchy problem for abstract parabolic equation and anisotropic elliptic equations are established.