In the present paper is given the analogue of the maximum principle for a scalar, linear eliptic equation, in coercive case (Lemma 1.4). The result is applied to locate the set of coincidence in the classical problem of Signorini for some concrete cases (Corollaries 1.6-1.8) and also, for the formulation of the maximum principle for the same problem (Theorem 1.5). An implicit Signorini problem was studied earlier by Bensoussan and Lions. They investigated the mentioned problem, proved existence, but the uniqueness result was still open. From the above mentioned results are derived uniqueness of a solution under asserted conditions. If some of asserted conditions is missing, the existence might fail; in particular, there are found a system of data, under which the problem has no solution at all. Next we state more general Siniorini's Implicit problem. In some cases, there is proved uniqueness of solution and is given a sufficient condition of solvability of the problem (Theorem 3.1). Further, is consider the implicit Signorini problem in elasticity with the Diriclet and the Neumann boundary conditions (Problem (4.20)-(4.21)). Existence of solution and, in some cases, also uniqueness is proved (Theorem 4.4). In general, uniqueness of solution, can equivalently be reduced to some assumption, similar to ``maximum principle'' (Lemma 1.4), of the theory of elasticity.
Mathematics Subject Classification: 35J20, 35J50.
Key words and phrases: An impicit signorini problem, coincidence set, coercivity property, nonhomogeneus body, a rigid fram.