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In some recent work in stochastic optimization with partial observation occurring in quality control problems, Anderson and Friedman [1], [2] have shown that the optimal cost can be determined as a solution of the quasi variational inequality  \[    M w ( p ) + f ( p ) ≥  0 , w ( p ) ≤  ψ  ( p ; w ) , ( M w ( p ) + f ( p ) ) ( w ( p ) −  ψ  ( p ; w ) ) = 0 \begin {gathered} Mw\left ( p \right )\, + \,f\left ( p \right )\, \geq \,0,\,w\left ( p \right )\, \leq \,\psi \left ( {p;\,w} \right ), \left ( {Mw\left ( p \right )\, + \,f\left ( p \right )} \right )\left ( {w\left ( p \right )\, - \,\psi \left ( {p;\,w} \right )} \right )\, = \,0 \end {gathered}     \]  in the simplex      p i &gt; 0 {p_i}\, &gt; \,0      ,      ∑  i = 1 n p i = 1 \sum \nolimits _{i\, = \,1}^n {{p_i}\, = \,1}      . Here  f ,      ψ  \psi       are given functions of  p ,      ψ  \psi       is a functional of  w , and  M  is a given elliptic operator degenerating on the boundary. This system has a unique solution when  M  does not degenerate in the interior of the simplex. The aim of this paper is to study the free boundary, that is, the boundary of the set where      w ( p ) &gt; ψ  ( p ; w ) w\left ( p \right )\, &gt; \,\psi \left ( {p;\,w} \right )      .

On the free boundary of a quasivariational inequality arising in a problem of quality control