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An \(L^ s\)-estimate for the gradient of solutions of some nonlinear unilateral problems. (English) Zbl 0599.49009

Let \(\Omega\) be a bounded domain in \(R^ N\) with sufficiently smooth boundary and let \(H_ 0^{1,p}(\Omega)\) for \(p>1\) be the standard Sobolev space. Consider the problem of finding \(u\in H_ 0^{1,p}(\Omega)\), \(u\geq \psi\) a.e. on \(\Omega\) such that \[ (1)\quad (Au-F,v-u)\geq 0,\quad for\quad all\quad v\in H_ 0^{1,p}(\Omega),\quad v\geq \psi \quad a.e.\quad in\quad \Omega, \] where \(\psi\), F are given functions and A is a differential operator. The author derives \(L^ s\)- estimates for the gradient of solutions to the nonlinear unilateral problem (1) for some \(s>p\). It is shown that this regularity result can be used to prove continuous dependence of the solution on the obstacle and its uniqueness in some quasilinear problems.
Reviewer: M.A.Noor

MSC:

49J40 Variational inequalities
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
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