We consider a total variation type energy which measures the jump discontinuities different from usual total variation energy. Such a type of energy is obtained as a singular limit of the Kobayashi-Warren-Carter energy with minimization with respect to the order parameter. We consider the Rudin-Osher-Fatemi type energy by replacing relaxation term by this type of total variation energy. We show that all minimizers are piecewise constant if the data function in the fidelity term is continuous in one-dimensional setting. Moreover, the number of jumps is bounded by an explicit constant involving a constant related to the fidelity. This is quite different from conventional Rudin-Osher-Fatemi energy where a minimizer has no jumps if the data has no jumps. Our results give an upper bound of the number of segments in a segmentation problem. If the function in fidelity is monotone, we have a shaper upper bound of the number of segments. We compare our results with segmentation by Mumford-Shah energy as well as by total variation energy. The existence of a minimizer is guaranteed in multi-dimensional setting when the data is bounded. We also prove that minimizers may not be unique. This talk is based on my recent joint work with A. Kubo (Hokkaido University), H. Kuroda (Hokkaido University), J. Okamoto (Meiji University) and K. Sakakibara (Kanazawa University).

报告人简介：Yoshikazu Giga（儀我美一），日本东京大学大学院数理科学研究科特任教授。曾先后任职于北海道大学、名古屋大学，2004年起任教于东京大学。长期从事非线性偏微分方程及其应用研究，在非线性扩散方程、几何演化方程、纳维-斯托克斯方程、界面运动及黏性解理论等方向取得了重要成果，是国际偏微分方程与几何分析领域的著名学者。

曾于2018年受邀在国际数学家大会（ICM）作45分钟特邀报告。曾获日本数学会秋季奖（小平邦彦奖）、井上科学奖、日产科学奖等学术荣誉，2010年获日本政府授予紫绶褒章，并当选美国数学会（AMS）会士、日本工业与应用数学学会（JSIAM）会士。