A remark on a system of inequalities with bilateral obstacles

Abstract

up+‘(x) k s up(x) I up+l(x) + K, x E Sz APUP = fP if up+‘(x) k < up(x) < up+‘(x) + K, x E Q APuP 5 fP if up(x) = up+l(x) + K, x E l2 (1.1) APuP L f if up(x) = up+‘(x) k, x E Q u”(x) = 0 onaQ,p= 1,2 ,..., m. where p ranges over (1, . . . , m) and urn+’ represents u’, is dealt with by the second author in [5], and the existence of a weak solution of (1.1) is established under the assumption: AP’s have sufficiently large zeroth order coefficients. (1.2) The proof of the existence result in [5] is based on the IV’*” estimate, independent of E, for the solution u, = 124:. . . . , u,“) of the following approximate system: APu, + &(u,p uf+l K) &(u;+~ k uf) = f in R, (1.3) u,” = 0 0naQ,p = 1,2 ,..., m, where p ranges over ( 1, . . . , ml, P, is the so-called penalty function and uF+’ represents u:. The gradient estimate for solutions U: of (1.3) in [5] is obtained by using Bernstein’s method, and assumption (1.2) is solely employed to make it possible. In addition, it should be remarked that standard variants of the Bernstein trick like in [2] and [4] which are useful to obtain the IV**” estimate for solutions of the system of variational inequalities with unilateral constraints without (1.2) do not work in our case.