Nonlocal problems in thin domains

摘要

In this paper we consider nonlocal problems in thin domains. First, we deal with a nonlocal Neumann problem, that is, we study the behavior of the solutions to f(x) = integral(Omega 1x Omega 2) J(is an element of)(x-y) (u(is an element of)(y) - u(is an element of)(x))dy with J(is an element of)(z) = J(z(1), is an element of z(2)) and Omega = Omega(1) x Omega(2) subset of R-N = R-N1 x R-N2 a bounded domain. We find that there is a limit problem, that is, we show that u(is an element of) -> u(0) as is an element of -> 0 in Omega and this limit function verifies integral(Omega 2) f(x(1), x(2))dx(2) = |Omega(2)| integral(Omega 1) J(x(1)-y(1),0) (U-0(y(1)) - U-0(x(1)))dy(1), with U-0(x(1)) = integral(Omega 2) u(0)(x(1), x(2))dx(2). In addition, we deal with a double limit when we add to this model a restate in the kernel with a parameter that controls the size of the support of J. We show that this double limit exhibits some interesting features.