𝐶₀-semigroups generated by second order differential operators with general Wentzell boundary conditions
Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli · Proceedings of the American Mathematical Society · 2000
Let us consider the operator A ~ u ( x ) = ϕ ( x , u ′ ( x ) ) u ( x ) , \widetilde {A}u(x)=\phi (x,u’(x))u(x), where ϕ \phi is positive and continuous in ( 0 , 1 ) × R (0,1)\times \mathbf {R} and A ~ \widetilde {A} is equipped with the so-called generalized Wentzell boundary condition which is of the form a A ~ u + b u ′ + c u = 0 a\widetilde {A} u+bu’+cu=0 at each boundary point, where ( a , b , c ) ≠ ( 0 , 0 , 0 ) . (a,b,c) eq (0,0,0). This class of boundary conditions strictly includes Dirichlet, Neumann and Robin conditions. Under suitable assumptions on ϕ \phi , we prove that A ~ \widetilde {A} generates a positive