Agmon Coerciveness and the Analysis of Operators with Formally Positive Integro-Differential Forms

For Sobolev spaces in Lipschitz domains with no imposed boundary conditions, the Aronszajn-Smith theorem algebraically characterizes coercive formally positive integro-differential quadratic forms. Recently, linear elliptic differential operators with formally positive forms have been constructed with the property that no formally positive forms for these operators can be coercive in any bounded domain. In the present article 4th order operators of this kind are shown by perturbation to have coercive forms that are (necessarily) algebraically indefinite. The perturbation here from noncoercive formally positive forms to coercive algebraically indefinite forms requires Agmon's characterization of coerciveness in smoother domains than Lipschitz.