AbstractThe aim of this article is to provide a functional analytical framework for defining the fractional powers{A^{s}} for {-1<s<1} of maximal monotone (possibly multivalued and nonlinear) operators A in Hilbert spaces.
We investigate the semigroup {\{e^{-A^{s}t}\}_{t\geq 0}} generated by {-A^{s}}, prove comparison principles and interpolations properties of {\{e^{-A^{s}t}\}_{t\geq 0}} in Lebesgue and Orlicz spaces.
We give sufficient conditions implying that {A^{s}} has a sub-differential structure.
These results extend earlier ones obtained in the case {s=1/2} for maximal monotone operators
[H. Brézis, Équations d’évolution du second ordre associées à des opérateurs monotones, Israel J. Math. 12 1972, 51–60],
[V. Barbu, A class of boundary problems for second order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 1972, 295–319],
[V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leiden, 1976],
[E. I. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 1986, 2, 514–543],
and the recent advances for linear operators A
obtained in
[L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian,
Comm. Partial Differential Equations 32 2007, 7–9, 1245–1260],
[P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 2010, 11, 2092–2122].