Degenerate elliptic operators in one dimension

作者：Robinson Derek W^*; Sikora Adam

来源：Journal of Evolution Equations, 2010, 10(4): 731-759.

DOI：10.1007/s00028-010-0068-9

Let H be the symmetric second-order differential operator on L(2)(R) with domain C(c)(infinity) (R) and action H(phi) = -(c phi')' where c is an element of W(loc)(1,2) (R) is a real function that is strictly positive on R\{0} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if nu = nu+ V nu- where nu+/-(x) = +/- integral(perpendicular to 1)(+/- x)c(-1) then H has a unique self-adjoint extension if and only if nu (is an element of) over bar L(2)(0, 1) and a unique submarkovian extension if and only if nu (is an element of) over bar L(infinity)(0, 1). In both cases, the corresponding semigroup leaves L(2)(0, infinity) and L(2)(-infinity, 0) invariant. In addition, we prove that for a general non-negative c is an element of W(loc)(1,infinity) (R) the corresponding operator H has a unique submarkovian extension.

• 出版日期2010-11