Ishii, Hitoshi; Yamada, Naoki A remark on a system of inequalities with bilateral obstacles. (English) Zbl 0718.49010 Nonlinear Anal., Theory Methods Appl. 13, No. 11, 1295-1301 (1989). The authors consider a system of elliptic variational inequalities with bilateral obstacles \[ \min \{\max \{A^ pu^ p-f^ p,u^ p- u^{p+1}-K\},u^{p+1}-u^ p-K\}=0\text{ in } \Omega,\quad u^ p=0\text{ on } \partial \Omega, \] with \(p=1,...,m\), where \(\Omega \subset {\mathbb{R}}^ N\) bounded domain, k, K are positive constants, \(f^ p\) smooth functions, and \(A^ p\) uniformly elliptic operators with smooth coefficients of the form \(A^ pv=-a^ p_{ij}(x)v_{x_ ix_ j}+b^ p_ i(x)v_{x_ i}+c^ p(x)v.\) In a previous work the second author had proved the existence of a weak solution of the system via \(W^{1,\infty}\) estimate under the assumptions: “\(A^ p\)’s have sufficiently large zeroth order coefficients”. In the present work they obtain, adopting Brandt’s method, the same \(W^{1,\infty}\) estimate but without the above assumption. Also they make some remarks about the advantage of this method. Reviewer: I.Athanasopoulos (Iraklion) Cited in 2 Documents MSC: 49J40 Variational inequalities 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35R35 Free boundary problems for PDEs Keywords:system of elliptic variational inequalities; bilateral obstacles; Brandt’s method PDFBibTeX XMLCite \textit{H. Ishii} and \textit{N. Yamada}, Nonlinear Anal., Theory Methods Appl. 13, No. 11, 1295--1301 (1989; Zbl 0718.49010) Full Text: DOI References: [1] Brandt, A., Interior estimates for second-order elliptic differential equations (or finite-difference) equations via the maximum principle, Israel J. Math., 7, 95-121 (1969) · Zbl 0177.37102 [2] Evans, L. C.; Friedman, A., Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Am. Math. Soc., 253, 365-389 (1979) · Zbl 0425.35046 [3] Ishii, H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s, Communs Pure Appl. Math, 42, 15-45 (1989) · Zbl 0645.35025 [4] Lions, P.-L., Resolution analytique des problemes de Bellman-Dirichlet, Acta Math., 146, 151-166 (1981) · Zbl 0467.49016 [5] Yamada, N., A system of elliptic variational inequalities associated with a stochastic switching game, Hiroshima math. J., 13, 109-132 (1983) · Zbl 0511.49004 [6] Yamada, N., Viscosity solutions for a system of elliptic inequalities with bilateral obstacles, Funkcialaj Ekvaciaj, 30, 417-425 (1987) · Zbl 0639.35032 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.