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Well-posedness of a nonlinear integro-differential problem and its rearranged formulation
Affiliation:1. HIVA–University of Louvain, Belgium;2. SHERPPA–Ghent University, Belgium;1. School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071, PR China;2. Institute of Applied Mathematics from Shandong Province, Qingdao, Shandong, 266071, PR China;1. Department of Civil Engineering, University of Ottawa, 161 Louis Pasteur, Ottawa, ON K1N6N5, Canada;2. Mathematics Department, Tulane University, 6823 St. Charles Ave., New Orleans, LA 70118, USA;3. Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA;1. Czech Geological Survey, Klárov 3, 118 21 Praha, Czech Republic;2. Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland;3. Poznań Radiocarbon Laboratory, Foundation of the Adam Mickiewicz University, Rubież 46, 61-612 Poznań, Poland;4. ITC - Faculty of Geo-Information Science and Earth Observation, University of Twente, 7500 AE Enschede, Netherlands;5. Geological Survey of Ethiopia, P.O. Box 2302, Addis Ababa, Ethiopia
Abstract:We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.
Keywords:Integro-differential equation  Existence and uniqueness  Neighborhood filters  Decreasing rearrangement  Denoising  Segmentation
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