GRADIENT BLOW-UP IN ZYGMUND SPACES FOR THE VERY WEAK SOLUTION OF A LINEAR ELLIPTIC EQUATION

It is known that the very weak solution of -integral(Omega)u Delta phi dx = integral(Omega)f phi dx, for all phi is an element of C-2((Omega) over bar), phi = 0 on partial derivative Omega, u is an element of L-1 (Omega) has its gradient in L-1 (Omega) whenever f is an element of L-1 (Omega; delta(1+ vertical bar Ln delta vertical bar), delta(x) being the distance of x is an element of Omega to the boundary. In this paper, we show that if f >= 0 is not in this weighted space L-1 (Omega; delta(1+ vertical bar Ln delta vertical bar)), then its gradient blows up in L (log L) at least. Moreover, we show that there exist a domain Omega of class C-infinity and a function f is an element of L-+(1) (Omega, delta) such that the associated very weak solution has its gradient being non integrable on Omega.