Sbordone, Carlo Grand Sobolev spaces and their application to variational problems. (English) Zbl 0915.46030 Matematiche 51, No. 2, 335-347 (1996). 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) Cited in 31 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49J40 Variational inequalities Keywords:variational problems; elliptic equation in divergence form; grand Sobolev space; embeddings; limiting space; Jacobian PDFBibTeX XMLCite \textit{C. Sbordone}, Matematiche 51, No. 2, 335--347 (1996; Zbl 0915.46030)