$C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions

Abstract

Let us consider the operator Au(x) = q(x,u'(x))u(x), where q is positive and continuous in (0,1) x R and A is equipped with the so-called generalized Wentzell boundary condition which is of the form aAu+bu'+cu = 0 at each boundary point, where (a, b, c) $4 (0,0,0). This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions. Under suitable assumptions on q, we prove that A generates a positive Cosemigroup on C[0, 1] and, hence, many previous (linear or nonlinear) results are extended substantially. INTRODUCTION In analysis, the boundary conditions associated with a second order (linear or nonlinear) elliptic differential operator usually involve the function and its first derivative (including Dirichlet, Neumann and Robin conditions). In Markov process theory, following the work of A.D. Wentzell [9], it was recognized that it is natural to include boundary conditions involving the operator itself. Thus, if A denotes an elliptic operator, the elliptic problem Au(x)-Au(x) = h(x) or the parabolic problem -Au= O a9t for x E Q C R', t > 0, is said to be equipped with the Wentzell boundary condition if one demands Au(x) = 0 for x E (9 (and any t > 0 in the parabolic case). The usual boundary condition can be written as