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

HTML articles powered by AMS MathViewer

- by Avner Friedman PDF
- Trans. Amer. Math. Soc.
**246**(1978), 95-110 Request permission

## Abstract:

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 \[ \begin {gathered} Mw\left ( p \right ) + f\left ( p \right ) \geq 0, w\left ( p \right ) \leq \psi \left ( {p; w} \right ), \left ( {Mw\left ( p \right ) + f\left ( p \right )} \right )\left ( {w\left ( p \right ) - \psi \left ( {p; w} \right )} \right ) = 0 \end {gathered} \] in the simplex ${p_i} > 0$, $\sum \nolimits _{i = 1}^n {{p_i} = 1}$. Here*f*, $\psi$ are given functions of

*p*, $\psi$ 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 $w\left ( p \right ) < \psi \left ( {p; w} \right )$.

## References

- Robert F. Anderson and Avner Friedman,
*A quality control problem and quasi-variational inequalities*, Arch. Rational Mech. Anal.**63**(1976/77), no. 3, 205–252. MR**456857**, DOI 10.1007/BF00251581 - Robert F. Anderson and Avner Friedman,
*Multidimensional quality control problems and quasivariational inequalities*, Trans. Amer. Math. Soc.**246**(1978), 31–76. MR**515529**, DOI 10.1090/S0002-9947-1978-0515529-8 - Alain Bensoussan, Haïm Brézis, and Avner Friedman,
*Estimates on the free boundary for quasi variational inequalities*, Comm. Partial Differential Equations**2**(1977), no. 3, 297–321. MR**473509**, DOI 10.1080/03605307708820032 - Alain Bensoussan and Avner Friedman,
*On the support of the solution of a system of quasivariational inequalities*, J. Math. Anal. Appl.**65**(1978), no. 3, 660–674. MR**510477**, DOI 10.1016/0022-247X(78)90170-1
A. Bensoussan and J.-L. Lions, - H. Brezis,
*Solutions of variational inequalities, with compact support*, Uspehi Mat. Nauk**29**(1974), no. 2 (176), 103–108 (Russian). Translated from the English by Ju. A. Dubinskiĭ; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. MR**0481460** - Luis A. Caffarelli,
*The regularity of free boundaries in higher dimensions*, Acta Math.**139**(1977), no. 3-4, 155–184. MR**454350**, DOI 10.1007/BF02392236 - L. A. Caffarelli and N. M. Rivière,
*Smoothness and analyticity of free boundaries in variational inequalities*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**3**(1976), no. 2, 289–310. MR**412940**

*Contrôle impulsionnel et temps d’arrêt inéquations variationnelles et quasi-variationnelles d’évolution*, Cahiers de mathématiques de la décision, no. 7523, Université Paris 9, Dauphine, 1975.

## Additional Information

- © Copyright 1978 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**246**(1978), 95-110 - MSC: Primary 93E20; Secondary 49A29, 62N10
- DOI: https://doi.org/10.1090/S0002-9947-1978-0515531-6
- MathSciNet review: 515531