Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces (Q631308)

The authors consider the ordinary differential operator \[ A_n u(x) = (-1)^{n + 1} \alpha(x) u^{2n}(x) \quad (0 \leq x \leq 1), \tag{1} \] where \(\alpha(x)\) is smooth, positive in \((0, 1)\) and vanishes at \(0, 1;\) the main example in mind is \(\alpha(x) = x^j(1 - x)^j,\) for which function the operator is called \(B_{n, j}\) \((B_n\) when \(j = n)\). In previous work, the authors have shown that \(B_1,\) with domain \(D(B_1) = \{u \in H_0^1(0, 1) \cap H^2_{loc}(0, 1): B_1 u \in H_0^1(0, 1)\}\), is selfadjoint and nonpositive in \(H_0^1(0, 1)\). The result is extended here to general operators \(A_n\) with domains in \(H_0^n(0, 1)\) with \(\alpha(\cdot) \in H_0^n(0, 1)\) satisfying additional boundedness and integrability conditions with respect to \(x^{2n}(1 - x)^{2n}\). These conditions are satisfied, for instance, by \(\alpha(x) = x^j(1 - x)^j\) with \(|j - n| < 1/2\).