Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces

Abstract

Let us consider the operator $A_n u$:=$(-1)^{n+1} \alpha (x) u^{(2n)}$ 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 $n\in\N$, $\alpha\in H^n_0(0,1)$, $\alpha (x)>0$ 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.