Selfadjointness of degenerate elliptic operators on higher order sobolev spaces


Let us consider the operator A nu:= (-1) n+ 1α(x)u (2n) on H 0n(0, 1) with domain D(A n):= {u ∈ H 0n(0,1) ∩ H loc2n(0, 1): A nu ∈ H 0n(0,1)}, where n ∈ N, α ∈ H 0n(0, 1), α(x) > 0 in (0, 1). Under additional boundedness and integrability conditions on α with respect to x 2n(1-x) 2n, we prove that (An, D(An)) is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on H 0n(0, 1). Analyticity results are also proved in Hn(0,1). In particular, all results work well when α(x) =x j (1-x) j for |j - n| < 1/2. Hardy type inequalities are also obtained.

Publication Title

Discrete and Continuous Dynamical Systems - Series S