Proceedings of the Royal Society of Edinburgh, 102A, 37-48, 1986 A remark on comparison results via symmetrization A. Alvino Istituto di Matematica, Universita di Napoli, Via Mezzocannone, 8, 80134 Naples, Italy P. L. Lions CEREMADE, Universite Paris IX-Dauphine, PL de Lattre de Tassigny, 75775 Paris Cedex 16, France and G. Trombetti Istituto di Matematica, Universita di Napoli, Via Mezzocannone, 8, 80134 Naples, Italy (MS received 20 September 1984. Revised MS received 2 April 1985) Synopsis In this paper, we study the converse of comparison results for solutions to linear second-order elliptic equations. Namely, in the inequalities proved by G. Talenti and others, we study the case of equality and prove that "equalities are achieved only in the spherical situation". We also present some applications of these results to semilinear elliptic equations. Introduction Let us first give the basic setting of this paper. We consider the Dirichlet problem for linear, second-order, elliptic equations in divergence form: - (Oi/iOx,. = / in n, u = 0 on dfl, (1) where we use the standard convention on repeated subscripts and Cl is a given bounded (or with finite measure) domain in IR N , / is a given function and a y are bounded measurable functions satisfying a.e. i n a (2) For any measurable function <p, we denote by <p* the decreasing rearrangement of <p into [0, +°°[, i.e. <p*(s) = inf{t S 0 ; n ( 0 ^ s } (3) with /x = /x<p given by /x(t) = meas (|<p(x)|> t). Then the (Schwarz) symmetrization—or spherical rearrangement—of <p, which we denote by <p # , is given by N ) (4)