Generalized versus classical normal derivative

Lucas Fresse, Viorica V. Motreanu


Given a bounded domain with Lipschitz boundary, the general Green formula permits to justify that the weak solutions of a Neumann elliptic problem satisfy the Neumann boundary condition in a weak sense. The formula involves a generalized normal derivative. We prove a general result which establishes that the generalized normal derivative of an operator coincides with the classical one, provided that the operator is continuous. This result allows to deduce that, under usual regularity assumptions, the weak solutions of a Neumann problem satisfy the Neumann boundary condition in the classical sense. This information is necessary in particular for applying the strong maximum principle.


Lipschitz domain; normal derivative; Green formula; generalized normal derivative; Neumann problem; strong maximum principle

Full Text:



Brezis, H., Analyse Fonctionnelle, Masson, Paris, 1983.

Casas, E., Fernandez, L.A., A Green's formula for quasilinear elliptic operators, J. Math. Anal. Appl., 142 (1989), no. 1, 62-73.

Evans, L.C., Gariepy, R.F., Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

Fresse, L., Motreanu, V.V., Axiomatic Moser iteration technique, submitted.

Grisvard, P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24, Pitman, Boston, MA, 1985.

Kenmochi, N., Pseudomonotone operators and nonlinear elliptic boundary value problems, J. Math. Soc. Japan, 27(1975), no. 1, 121-149.

Lieberman, G.M., Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12(1988), 1203-1219.

Motreanu, D., Motreanu, V.V., Papageorgiou, N., Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems, Springer, New York, 2014.

Vazquez, J.L., A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12(1984), no. 3, 191-202.



  • There are currently no refbacks.