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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- NikodĂ˝m 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)

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