Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations （Mu）＇（t） = Au（t） ...Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations （Mu）＇（t） = Au（t） ＋ f（t） for t ∈ R in HSlder continuous function spaces C^α（R; X） by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D（A） D（M）.展开更多
Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in o...Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.展开更多
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant No. 11171172） and the Specialized Research Fund for the Doctoral Program of Higher Education （No. 20120002110044）.
文摘Using known operator-valued Fourier multiplier results on vectorvalued HSlder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations （Mu）＇（t） = Au（t） ＋ f（t） for t ∈ R in HSlder continuous function spaces C^α（R; X） by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D（A） D（M）.
文摘Let Q（x） be a nonnegative definite, symmetric matrix such that √Q（X） is Lipschitz con- tinuous. Given a real-valued function b（x） and a weak solution u（x） of div（QVu） = b, we find sufficient conditions in order that √Qu has some first order smoothness. Specifically, if is a bounded open set in Rn, we study when the components of vVu belong to the first order Sobolev space W1＇2（Ω） defined by Sawyer and Wheeden. Alternately we study when each of n first order Lipschitz vector field derivatives Xiu has some first order smoothness if u is a weak solution in Ω of ^-^-1 X^Xiu ＋ b = O. We do not assume that {Xi}is a HSrmander collection of vector fields in ~. The results signal ones for more general equations.
