Optimal regular differential operators with variable coefficients and application

In this paper, maximal regularity properties for linear and nonlinear high order elliptic differential-operator equations with VMO coefficients are studied. For linear case, the uniform coercivity properties of parameter dependent boundary value problems is obtained in L^{p} spaces. Then, the existence and uniqueness of strong solution of the boundary value problem for high order nonlinear equation is established. In application, the maximal regularity properties of the anisotropic elliptic equation and system of equations with VMO coefficients are derived.