**Avtandil Gachechiladze**

## A maximum principle and the implicit signorini problem

**abstract:**

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.