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Existence and stability results in the $$L^1$$ theory of optimal transportation. (English) Zbl 1065.49026
Caffarelli, Luis A. (ed.) et al., Optimal transportation and applications. Lectures given at the C.I.M.E. summer school, Martina Franca, Italy, September 2–8, 2001. Berlin: Springer (ISBN 3-540-40192-X/pbk). Lect. Notes Math. 1813, 123-160 (2003).
The earliest version of the optimal transportation problem was formulated by Monge in 1781. In it, two distributions of matter in $${\mathbb R}^2$$ are given by two nonnegative Borel functions $$h_0(x), h_1(x)$$ with equal integrals over $${\mathbb R}^2.$$ A transport $$t : {\mathbb R}^2 \to {\mathbb R}^2$$ is a Borel map satisfying the local balance of mass condition $\int_{t^{-1}(e)} h_0(x) dx = \int_e h_1(x) dx \quad (e = \text{Borel set} \subseteq {\mathbb R}^2) \, .$ The Monge problem is that of minimizing the transportation work, $\min \bigg\{ \int_{{\mathbb R}^2} | t(x) - x| h_0(x) dx : t \;\text{transport} \bigg\}.$ Subsequently, the problem has been generalized in various directions, where $${\mathbb R}^2$$ becomes $${\mathbb R}^n$$ or a more general space, the $$h_j$$ are measures and the Euclidean distance $$| x - y|$$ is replaced by a function $$c(x, y).$$ Weak solutions (called plannings) of the transportation problem were defined by Kantorovich in 1942.
Under conditions that include $$c(x, y) = h(x - y)$$ with $$h$$ strictly convex it can be shown that a weak solution of the transportation problem induces a transport $$t(x).$$ Without the strict convexity condition this is not the case in general, and information on $$t(x)$$ is lost. The main results in this paper are on approximation on the cost function $$c(x, y) = \| x - y\|$$ by $$c_\epsilon(x, y) = \| x - y\| ^{1 + \epsilon}$$ as $$\epsilon \to 0$$ $$(\| \cdot\|$$ a norm in $${\mathbb R}^n$$), and include existence and stability theorems. There is also much additional information (accessible to the nonspecialist) on the optimal transportation problem.
For the entire collection see [Zbl 1013.00028].

##### MSC:
 49Q20 Variational problems in a geometric measure-theoretic setting 49J45 Methods involving semicontinuity and convergence; relaxation