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Grand Sobolev spaces and their application to variational problems. (English) Zbl 0915.46030

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\). Let \(1<q< \infty\). Then the grand space \(L_{q)}(\Omega)\) consists of all \(v\in L_1(\Omega)\) such that \[ \| v\mid L_{q)}(\Omega)\|= \sup_{\varepsilon>0} \Biggl(\varepsilon \int_\Omega | v(x)|^{q-\varepsilon} dx\Biggr)^{\frac{1}{q-\varepsilon}}< \infty \] and the grand Sobolev space \(W_{q)}^1(\Omega)\) consists of all \(u\in L_q(\Omega)\) with \(\nabla u\in L_{q)}(\Omega)\). The paper deals with embeddings, especially for the limiting space \(W_{n)}^1(\Omega)\), and the behaviour of the Jacobian in the vector-valued case, and applications to elliptic equations in divergence form.
Reviewer: H.Triebel (Jena)

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J40 Variational inequalities
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