Some problems in differential operators - Keller, G; Keller, R. G. (1977)
We consider a formally self-adjoint elliptic differential operator in ℝn, denoted by τ. T0 and T are operators given by τ with specific domains. We determine conditions under which T0 is essentially self-adjoint, introducing the topic by means of a brief historical survey of some results in this field. In Part I, we consider an operator of order 4, and in Part II, we generalise the results obtained there to ones for an operator of order 2m. Thus, the two parts run parallel. In Chapter 1, we determine the domain of T0*, denoted by D(T0*), where T0* denotes the adjoint of T0, and introduce operators T0 and T which are modifications of T0 and T. In Chapter 2, we use a theorem of Schechter to give conditions under which T0 is essentially self-adjoint. Working with the operator T, in Chapter 3 ve show that we can approximate functions u in D(T0*) by a particular sequence of test-functions, which enables us to derive an identity involving u, Tu and the coefficient functions of the operator concerned. In Chapter 4, we determine an upper bound for the integral of a function involving a derivative of u in D(T0*) whose order is half the order of the operator concerned, and we use the identity from the previous chapter to reformulate this upper bound. In Chapter 5, we give conditions which are sufficient for the essential self-adjointness of T0. In the main theorem itself, the major step is the derivation of the integral of the function involving the particular derivative of u in D(T0*) whose order is half the order of the operator concerned, referred to above, itself as a term of an upper bound of an integral we wish to estimate. Hence, we can employ the upper bound from Chapter 4. This "sandwiching" technique is basic to the approach we have adopted. We conclude with a brief discussion of the operators we considered, and restate the examples of operators which we showed to be essentially self-adjoint.