Abstract
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 X_{0}, with constant domain D(A(t))=X_{1} for every t∈ (0,T). Maximal L^{p}regularity means that for all f∈ L^{p}(0,T;X_{0}), the solution of the above evolution problem is such that u', Au are both in L^{p}(0,T;X_{0}). 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 LopatinskiiShapiro type.
We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X_{0}, with constant domain D(A(t))=X_{1} for every t∈ (0,T). Maximal L^{p}regularity means that for all f∈ L^{p}(0,T;X_{0}), the solution of the above evolution problem is such that u', Au are both in L^{p}(0,T;X_{0}). 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 LopatinskiiShapiro type.
Original language  English 

Awarding Institution 

Supervisors/Advisors 

Award date  31 Mar 2017 
Print ISBNs  9789402805543 
DOIs  
Publication status  Published  2017 
Keywords
 Integral operators
 maximal Lpregularity
 functional calculus,
 selliptic and parabolic equations
 Apweights
 Rboundedness
 extrapolation
 quasilinear PDE
 Fourier multipliers
 the Lopatinskii–Shapiro condition
 mixednorm