Boundary oscillations and nonlinear boundary conditions
José M. Arrieta, Simone Mazzini Bruschi · Comptes Rendus Mathématique · 2006
We study how oscillations in the boundary of a domain affect the behavior of solutions of elliptic equations with nonlinear boundary conditions of the type ∂ u ∂ n + g ( x , u ) = 0 . We show that there exists a function γ defined on the boundary, that depends on the oscillations at the boundary, such that, if γ is a bounded function, then, for all nonlinearities g , the limiting boundary condition is given by ∂ u ∂ n + γ ( x ) g ( x , u ) = 0 (Theorem 2.1, Case 1). Moreover, if g is dissipative and γ ≡ ∞ then we obtain a Dirichlet boundary condition (Theorem 2.1, Case 2).