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Convergence in \({\mathcal D}'\) and in \(L^ 1\) under strict convexity. (English) Zbl 0813.49016
Lions, Jacques-Louis (ed.) et al., Boundary value problems for partial differential equations and applications. Dedicated to Enrico Magenes on the occasion of his 70th birthday. Paris: Masson. Res. Notes Appl. Math. 29, 43-52 (1993).
Let \(\Omega\subset \mathbb{R}^ n\) be a bounded open set and let \((u_ n)\) be a sequence in \(L^ 1(\Omega; \mathbb{R}^ n)\) which converges “weakly” to some limit \(u\in L^ 1(\Omega; \mathbb{R}^ n)\). Let \({\mathcal J}: \mathbb{R}^ n\to \mathbb{R}\) be a convex function such that \[ \limsup_{n\to \infty} \int_ \Omega {\mathcal J}(u_ n)\leq \int_ \Omega {\mathcal J}(u).\tag{1} \] The assumption that \((u_ n)\) converges to \(u\) weakly in \(L^ 1\) (that is, for the weak \(\sigma(L^ 1, L^ \infty)\) topology) is a very restrictive assumption and it is desirable to replace it by the much weaker and more natural assumption that \((u_ n)\) converges to \(u\) in the sense of distributions \[ u_ n\to u\quad\text{in}\quad D'(\Omega; \mathbb{R}^ n).\tag{2} \] The main theorem of this paper is the following:
Let \((u_ n)\) be a sequence in \(L^ 1(\Omega; \mathbb{R}^ n)\) and let \(u\in L^ 1(\Omega; \mathbb{R}^ n)\) be such that (1) and (2) hold. Assume that \(\mathcal J\) is strictly convex. Then \(u_ n\to u\) strongly in \(L^ 1_{\text{loc}}(\Omega; \mathbb{R}^ n)\). If, in addition, we suppose that \(\lim_{| t|\to \infty} {\mathcal J}(t)= +\infty\), then \(u_ n\to u\) strongly in \(L^ 1(\Omega; \mathbb{R}^ n)\).
For the entire collection see [Zbl 0782.00097].

49J45 Methods involving semicontinuity and convergence; relaxation