Homogenization for the p-Laplacian in an n-dimensional domain perforated by very thin cavities with a nonlinear boundary condition on their Boundary in the case p = n

摘要

We investigate the asymptotic behavior, as epsilon -> 0, of the solution u (epsilon) to the boundary value problem for the equation -I" (p) u (epsilon) = f in a domain Omega(epsilon) aS, a"e (n) perforated by very thin arbitrarily shaped cavities separated by an O(epsilon) distance in the case of p = n a parts per thousand yen 3 with a nonlinear third boundary condition of the form specified on their boundary, where nu is the outward unit normal vector on the boundary of the cavities. The adsorption coefficient beta(epsilon) and the perforation radius a (epsilon) satisfy conditions that are critical to the given problem. A homogenized model is constructed, and the solutions u (epsilon) are proved to converge weakly, as epsilon -> 0, to the solution of the homogenized problem.