OnLp-Contractivity of Semigroups Generated by Linear Partial Differential Operators

This paper is devoted to the study of contraction semigroups generated by linear partial differential operators. It is shown that linear partial differential operators of order higher than two cannot generate contraction semigroups on (Lp)Nforp∈[1, ∞) unlessp=2. Ifp>1 and theLp-dissipativity criterion is restricted to the cone of nonnegative functions for differential operators with real-valued coefficients, it is proven that the criterion still fails for operators of order higher than two, except for some fourth order operators if 3/2⩽p⩽3. A class of such fourth order operators is also presented.