Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces

Let us consider the operator A_nu:=(-1)^{n+1}\alpha (x)u^(2n) on H^n_0(0,1) with domain D(A_n):={u\in H^n_0(0,1)\cap H^{2n}_{loc}(0,1): A_n u\in H^n_0(0,1)}, where n\in\bold N, \alpha\in H^n_0(0,1), \alpha (x)>0 in (0,1).Under additional boundedness and integrability conditions on \alpha with respect to x^{2n} (1-x)^{2n}, we prove that (A_n, D(A_n)) is nonpositive and selfadjoint, thus it generates a cosine function, hence an analytic semigroup in the right half plane on H^n_0(0,1). Analyticity results are also proved in H^n(0,1). In particular, all results work well when \alpha (x)=x^j (1-x)^j, for |j-n|<1/2. Hardy type inequalities are also obtained.