# zbMATH — the first resource for mathematics

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)

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation 46A50 Compactness in topological linear spaces; angelic spaces, etc.
Full Text:
##### References:
  Ambrosio L.: A compactness theorem for a new class of functions of bounded variation. Boll. Un. Mat. Ital.3-B (1989) 857–881 · Zbl 0767.49001  Ambrosio L.: Existence theory for a new class of variational problems. Arch. Rat. Mech. Anal.111 (1990) 291–322 · Zbl 0711.49064  Ambrosio L.: Variational problems in SBV. Acta Applicandae Mathematicae17 (1989) 1–40 · Zbl 0697.49004  Ambrosio L., Dal Maso G.: A general chain rule for distributional derivatives. Proc. Am. Mat. Soc.108 (1990) 691–702 · Zbl 0685.49027  Carriero M., Leaci A., Tomarelli F.: Plastic free discontinuities and special bounded hessian. C.R. Acad. Sci. Paris.314 (1992) 595–600 · Zbl 0794.49011  De Giorgi E., Ambrosio L.: Un nuovo tipo di funzionale del Calcolo delle Variazioni. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Nat.82 (1988) 199–210  De Giorgi E., Carriero M., Leaci A.: Existence theorem for a minimum problem with free discontinuity set. Arch. for Rat. Mech. Anal.108 (1989) 195–218 · Zbl 0682.49002  Evans L.C., Cariepy R.F.: Lecture notes on measure theory and fine properties of functions. ERC Press, Boca Raton, 1992  Federer H.: Geometric measure theory. Springer, Berlin Heidelberg New York 1969 · Zbl 0176.00801  Giaquinta M, Modica G., Soucek J.: Area and the area formula (to appear) · Zbl 0421.49010  Lin F.H.: Variational problems with free interfaces. Calc. Var.1 (1993) 149–168 · Zbl 0794.49038  Mantegazza C.: Curvature varifolds with boundary (to appear) · Zbl 0865.49030  Morgan F.: Geometric Measure Theory – A beginner’s guide. Academic Press, Boston 1988 · Zbl 0671.49043  Mumford D., Shah J.: Optimal approximation by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.17 (1989) 577–685 · Zbl 0691.49036  Ossanna E.: SBV functions defined on currents (to appear) · Zbl 0877.49002  Simon L.: Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, Camberra 1983 · Zbl 0546.49019  Volpert A.I.: Spaces BV and quali-linear equations. Math. USSR Sb.17 (1969) 225–267  Volpert A.I., Hudjaev S.I.: Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics. Kluwer Academic Publishers, Dordrecht 1985
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.