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Existence of solutions of obstacle problems. (English) Zbl 0735.35067
The purpose of this important paper is to study the question concerning the existence of solutions to obstacle problems. The pointwise regularity of solutions to certain obstacle problems (variational inequalities) is investigated. The authors have shown that, under certain conditions (\(\Omega\) is a bounded nonempty open set of \(\mathbb{R}^ n\) and \(\psi\) is a real-valued function on \(\Omega\)), there exists \(u\in W^{1,\alpha}(\Omega)\), such that \[ u(x)\geq\psi(x) \tag{1} \] for \(x\) in a certain subset of \(\Omega\), \(u\) satisfies a given boundary condition and \[ \sum_{i=1}^ n\int_ \Omega a_ i(x,u(x),Du(x)){\partial\varphi \over\partial x_ i}(x)dx+\int_ \Omega b(x,u(x),Du(x))\varphi(x)dx\geq0 \tag{2} \] for all \(\varphi\) in a specified subspace of \(W_ 0^{1,\alpha}(\Omega)\) with (3) \(\varphi(x)\geq\psi(x)-u(x)\) for all \(x\) in a specified subset of \(\Omega\).
Main result: In the existence theorem an integral average procedure is used for extending the domain of Sobolev functions to quasi-all (in the sense of an appropriate capacity) \(\Omega\) and the authors require that (1) and (3) hold for quasi-all \(x\in\Omega\). This means that this work generalizes the work of those authors who require only that (1) and (3) hold for almost all \(x\in\Omega\).

MSC:
35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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