Fractional Powers of Monotone Operators in Hilbert Spaces

Abstract

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