Gary M. Lieberman

The conormal derivative problem for equations of variational type in nonsmooth domains

it is well known that elliptic boundary value problems in smooth domains have smooth solutions, but if the domain is, say, C 1 , the solutions need not be Lipschitz. Recently Korevaar has identified a class of Lipschitz domains, in which solutions of the capillary problem are Lipschitz assuming the contact angle relates correctly to the geometry of the domain. Lipschitz bounds for more general boundary value problems in the same class of domains are proved. Applications to variational inequalities are also considered

Two-dimensional nonlinear boundary value problems for elliptic equations

On considere la regularite limite des solutions du probleme aux valeurs limites non lineaire F(x,u,Du,D 2 u)=0 dans Ω, G(x,u,Du)=0 sur ∂Ω pour des domaines Ω a 2 dimensions. F est uniformement elliptique et G depend de Du. On demontre des estimations de continuite pour des derivees premieres et secondes de u sous des hypotheses faibles pour la regularite de F, G et Ω

Solvability of quasilinear elliptic equations with nonlinear boundary conditions. II

Abstract It is shown that solvability of the second order quasilinear elliptic equation Qu = 0 in Ω with first order nonlinear boundary condition Nu = 0 on ∂Ω can be inferred from appropriate estimates on solutions of the problem Qu = ƒ in Ω, Nu = 0 on ∂Ω as ƒ varies over a suitable function class. This result improves previous work of the author, where estimates were required on solutions of Qu = ƒ in Ω, Nu = ϑ on ∂Ω as (ƒ, ϑ) varies over some function space. The value of this improvement is demonstrated by some examples.