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Embedding theorems for elliptic equations in unbounded domains. (Italian. English summary) Zbl 0763.35028

Let \(\rho(x)=(1+| x|)/2\) defined on a smooth unbounded open set \(\Omega\) with bounded boundary. The authors study some weighted Sobolev spaces among which is \(W^ m_{\sigma}(\Omega)\) defined using the weight \(\rho^{2\sigma}\). As an application, the estimate \(| u|_{W^ 2_{\sigma}(\Omega)}\leq k (| Lu|_{L^ 2_{\sigma}(\Omega)}+| u|_{2,\Omega_ o})\) is obtained, where \(Lu\) is an elliptic linear differential operator satisfying the Cordes condition and \(\Omega_ o\) is a bounded open subset of \(\Omega\).

MSC:

35J25 Boundary value problems for second-order elliptic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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