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A remark on a system of inequalities with bilateral obstacles. (English) Zbl 0718.49010

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.

MSC:

49J40 Variational inequalities
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35R35 Free boundary problems for PDEs
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