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A new proof of the SBV compactness theorem. (English) Zbl 0837.49011
The paper studies a compactness theorem for functions belonging to the space \(\text{SBV}(\Omega)\), where \(\Omega\) denotes a bounded open set in \(\mathbb{R}^n\) and \(\text{SBV}(\Omega)\) is defined as follows: The space \(\text{BV}(\Omega)\) consists of measurable functions \(u\in L^1(\Omega)\) such that the derivatives \(Du= (D_1 u,\dots, D_n u)\) in distributional sense are of finite total variation in \(\Omega\). Let \(D^s u\), \(u\in \text{BV}(\Omega)\), denote the singular part of the Radon- Nikodým decomposition. Then \(D^s u\) is further decomposed as the sum \(Ju+ Cu\), where \(Ju\) denotes the restriction of \(D^s u\) to the jump set \(S_u\) and \(Cu\) the restriction to \(\Omega\backslash S_u\). Then the space \(\text{SBV}(\Omega)\) consists of \(u\in \text{BV}(\Omega)\) such that \(Cu= 0\).
The main result is stated as follows: Let \(\varphi: [0, \infty)\to [0, \infty]\) be a convex function such that \(\lim_{t\to \infty} \varphi(t)/t= \infty\). Let \(\{u_h\}\) be a sequence in \(\text{SBV}(\Omega)\) such that \(|u_h|_\infty+ \int_\Omega \varphi(|\nabla u_h|)dx\) is uniformly bounded from above. Then there is a subsequence \(\{v_k\}\) such that (i) it converges to some \(v\in \text{SBV}(\Omega)\) in \(L^1_{\text{loc}}(\Omega)\); (ii) \(\nabla v_k\) weakly converges to \(\nabla v\) in \(L^1(\Omega; \mathbb{R}^n)\); and (iii) \({\mathcal H}^{n- 1}(S_v)\leq \liminf {\mathcal H}^{n-1}(S_{v_k})\), where \({\mathcal H}^{n- 1}\) denotes the Hausdorff \((n-1)\)-dimensional measure in \(\mathbb{R}^n\). The proof of this result gives a simplification of a previous one by the same author.
Reviewer: T.Nambu (Kobe)

49J45 Methods involving semicontinuity and convergence; relaxation
46A50 Compactness in topological linear spaces; angelic spaces, etc.
Full Text: DOI
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