Existence Theorem for a Minimum Problem with Free Discontinuity Set E. DE GIORGI, M. CARRIERO • A. LEACI Communicated by E. GIUSTI Abstract We study the variational problem min{aflVul2dx+#o\rf lu--glqdx-k 2H"-'(KAO); K Q R n closed set, u C C*(f2 \ K)} where f2 is an open set in R n, n => 2, gELq(I2) AL~(s 1 <= q < +co, 0 < 2, # < -t-oo and H~_ I is the (n -- 1)-dimensional Hausdorff measure. 1. Introduction In this paper the following theorem is proved. Theoreml.1. Let nEN, n~2, let ~C=R n be an open set, 1 ~q~ ~co, 0 "( ~ ~ -}-cx~, 0 ~ I z ~ q-cx:~, g E L q ( ~ ) A L~(f2); then there ex&ts at least one pair (K, u) minimizing the functional G defined for every closed set K Q R" and for every u 6 Cl(K2 \ K) by G(K, u) = f IVul2 dx + /~ f lu -- glq dx + 2H,,_,(K A ~), s [2\K where Hn_ 1 is (n- 1)-dimensional Hausdorff measure. This theorem was announced under slightly more restrictive hypotheses, during the Meeting in honor of J.-L. LIONS held in Paris from 6 to 10 June 1988 (see [10]). It provides the beginning of a positive answer to a two-dimensional problem of image segmentation in Computer Vision Theory posed by D. MUM- FORD • J. SHAH in [21].