Maximal regularity for parabolic equations with measurable dependence on time and applications

The subject of this thesis is the study of maximal L^p-regularity of the Cauchy problem u'(t)+A(t)u(t)=f(t), t∈ (0,T), u(0)=x.
We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X₀, with constant domain D(A(t))=X₁ for every t∈ (0,T). Maximal L^p-regularity means that for all f∈ L^p(0,T;X₀), the solution of the above evolution problem  is such that u', Au are both in L^p(0,T;X₀).   In the first part of the thesis, we introduce a new operator theoretic approach to maximal L^p-regularity in the case the dependence t→A(t) is just measurable. The abstract method is then applied to concrete parabolic PDEs: we consider equations and systems of elliptic differential operators of even order, with coefficients measurable in the time variable and continuous in the space variables, and we show that they have maximal L^p-regularity on L^q(\R^d), for every p,q∈(1,∞). These results gives an alternative approach to several PDE results in the literature, where only the cases p=q or q≤p were considered. As a further example, we apply  our abstract approach also to higher order differential operators with general boundary conditions, on the half space, under the same assumptions.
The last part of this thesis is based on a different approach and it is devoted to the study of maximal L^p-regularity on L^q(\R^d₊) of an elliptic differential operator of higher order with coefficients in the class of vanishing mean oscillation both in the time and the space variables, and general boundary conditions of Lopatinskii-Shapiro type.