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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].

MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation