关于全变差非局部逼近的Gamma极限与逐点极限之间的差距
On the gap between Gamma-limit and pointwise limit for a non-local approximation of the total variation
摘要：
本文研究了由H。Brezis和H.- M. Nguyen在最近的一篇论文中说。通过二重积分定义逼近泛函，其中每对点按某种相互作用规律进行积分。在本文中，我们回答了两个公开的问题，有关的相互作用定律的伽玛极限的依赖性。在第一个结果中，我们表明，伽玛极限依赖于完整的相互作用定律的形状，而不仅仅是在原点附近的值。在第二个结果中，我们证明了在光滑函数上确实存在Gamma极限与逐点极限重合的相互作用律。关键的论点是，对于某些特殊类的相互作用律的伽玛极限的计算可以减少到研究适当的多变量最小问题的渐近行为。
英文摘要：
We consider the approximation of the total variation of a function by the family of non-local and non-convex functionals introduced by H. Brezis and H.-M. Nguyen in a recent paper. The approximating functionals are defined through double integrals in which every pair of points contributes according to some interaction law. In this paper we answer two open questions concerning the dependence of the Gamma-limit on the interaction law. In the first result, we show that the Gamma-limit depends on the full shape of the interaction law, and not only on the values in a neighborhood of the origin. In the second result, we show that there do exist interaction laws for which the Gamma-limit coincides with the pointwise limit on smooth functions. The key argument is that for some special classes of interaction laws the computation of the Gamma-limit can be reduced to studying the asymptotic behavior of suitable multi-variable minimum problems.
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