L~p-Maximal Regularity For Incomplete Second Order Cauchy Problems

Establishing L~p-maximal regularity for large classes of classical partial differential operators and systems does help to study regularity of their solutions and existence of nonlinear equations. In fact, frequently a partial differential equation can be transformed into an ordinary differential equation with values in an infinite dimensional space. Cauchy problem is cauchy style of partial differential equations. For these reasons much attendtion have been paid to L~p-maximal regularity of one order Cauchy problem since sixties of last century. In the ninties good results have been obtained. This paper is to study L~p-maximal regularity for imcomplete second order Cauchy problems, furthermore results from various sides have been obtained. Two criteria—multipler and functional calculus of L~p-maximal regularity are obtained, and the analyticity of the corresponding sine operator is considered. On the other hand, we introduce the connection of L~p-maximal regularity among imcomplete second order Cauchy problems, periodic problems, the corresponding one order Cauchy problems.In this paper we use multiplier characterization and operator-sum method, that is to say, L.Weis' operator-valued multiplier theorem, operator-sum theorem and operator-valued Marcinkiewicz multiplier theorem. Meanwhile, we use many methods, techniques and results from the functional analysis including semigroup of operators, functional calculs, operator-sum theorem, embeded theorem, UMD-space and so on. We use some results and means of one order Cauchy problem to obtain systematic results on L~p-maximal regularity of imcomplete second order Cauchy problems.The significance of this paper is not only to perfect the system on L~p-maximal regularity of imcomplete second order Cauchy problems, but also set a foundation for the posedness of quasilinear evolution equations and non-automous systems.