Analytic semigroups on C[0, 1] generated by some classes of second order differential operators

If alpha, beta is an element of C[0,1], alpha > 0 in (0, 1) and alpha(0) = 0 = alpha(1), we consider the second order differential operator on C[0,1] defined by Au := alpha u " + beta u', where D(A) may include Wentzell boundary conditions. Under integrability conditions involving root alpha and beta/root alpha, we prove the analyticity of the semigroup generated by (A, D(A)) on C-o[0, 1], C-pi[0, 1] and on C[0, 1], where C-o[0,1] := {u is an element of C[0,1]\u(0) = 0 = u(1)} and C-pi[0,1]:= {u is an element of C[0,1]\u(0) = u(1)}. We also prove different characterisations of D(A) related to some results in [1], where beta = 0, exhibiting peculiarities of Wentzell boundary conditions. Applications can be derived for the case alpha(x) := x(j)(1 - x)(j)(j greater than or equal to 1, x is an element of [0,1]) and beta(x) = x(k)(1 - x)(k) gamma(x) (k greater than or equal to j/2, x is an element of [0,1], gamma is an element of C[0,1]).