Minimization of an Isoperimetric Functional Perturbed by a Regular-Kernel Potential
DOI:
https://doi.org/10.15849/ijasca.180Keywords:
Classical perimeter, Riesz’s Rearrangement inequality, Shape optimization, Monte Carlo methodAbstract
In this paper, we investigate the minimization of the shape functional energy defined by the sum of the classical perimeter with a competing potential of a regular kernel, which is maximized by the ball. On the class of measurable subsets of the m-dimensional Euclidean space, which are of finite perimeter and have a fixed volume v, we prove the existence of a minimizer and we determine a condition on the kernel for which the ball becomes the minimizer of the considered problem. We then develop a MATLAB-based numerical algorithm to approximate the shape functional problem for the kernel exponential. The numerical results indicate the existence of a threshold area , which depends on the area and the regular kernel, such that below which the disk appears to minimize the shape functional among the considered shapes. Also, we augmented the considered shape functional by adding the Fraenkel asymmetry of E multiplied by , we prove that the ball of volume is a minimizer of the augmented shape functional.
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Copyright (c) 2026 Idriss OUSKHNIDCopyright © The Author(s).
Articles published in the International Journal of Advances in Soft Computing and its Applications (IJASCA) are licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
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