Fractional Powers of Monotone Operators in Hilbert Spaces

Abstract

Abstract The aim of this article is to provide a functional analytical framework for defining the fractional powers A s {A^{s}} for - 1 < s < 1 {-1<s<1} of maximal monotone (possibly multivalued and nonlinear) operators A in Hilbert spaces. We investigate the semigroup { e - A s ⁢ t } t ≥ 0 {\{e^{-A^{s}t}\}_{t\geq 0}} generated by - A s {-A^{s}} , prove comparison principles and interpolations properties of { e - A s ⁢ t } t ≥ 0 {\{e^{-A^{s}t}\}_{t\geq 0}} in Lebesgue and Orlicz spaces. We give sufficient conditions implying that A s {A^{s}} has a sub-differential structure. These results extend earlier ones obtained in the case s = 1 / 2 {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].