Nonlocal elliptic problems and applications

In this paper, the integral boundary value problems for differential-operator equations with principal variable coefficients are studied. Several conditions for the $L^{p}$-separability are given. Moreover, the sharp coercive estimates for resolvents of corresponding differential operators are shown. It is implied that these operators are positive and also are generators of analytic semigroups. Then, the existence and uniqueness of maximal regular solution to nonlinear abstract elliptic equations is derived. In application, maximal regularity properties of the abstract parabolic equation with variable coefficients and systems of elliptic equations are derived in mixed $L^{\mathbf{p}}$-spaces.