Geometric framework for gradient flow in shape optimization

Geometric framework for gradient flow in shape optimization Full article

Journal

Успехи математических наук
ISSN: 0042-1316

Output data

Year: 2026, Volume: 81, Number: 3(489), Pages: 51-150 Pages count : 100 DOI: 10.4213/rm10296

Tags

shape optimization, gradient flows, Kohn–Vogelius functional, Bernoulli problem

Authors

Plotnikov Pavel Igorevich ¹ , Sokolowski Jan ²

Affiliations

1	Lavrentyev Institute of Hydrodynamics of Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia
2	Institut Elie Cartan, Laboratoire de Mathematiques, Universite de Lorraine, Nancy, France

This paper explores gradient flow dynamical systems in the context of shape optimization problems, with a particular focus on the regularization of shape functionals. Two representative model problems are selected for this purpose: one from inverse problems and the other from free boundary problems. The first model is a single-measurement identification problem, which arises in electrical impedance tomography, especially in geophysical applications. The second is the Bernoulli free boundary problem, known for its broad range of applications. Throughout the paper, these problems serve as illustrative examples for fundamental methods and approaches in shape optimization theory. Variational formulations are developed for both problems. In the case of the inverse problem the formulation is based on minimizing the integral Kohn–Vogelius functional, with a regularization strategy involving the one-dimensional Willmore functional. For the free boundary problem an energy-type functional is considered, and an additional regularization using the M¨obius energy is introduced. In both cases, perimeter regularization is sufficient to ensure the existence of an optimal curve. However, stronger regularization is required to guarantee the convergence of the associated gradient flow dynamical systems. The paper revisits basic concepts from the differential geometry of curves in the Euclidean plane and derives explicit formulae for the gradients of both the Kohn–Vogelius and Bernoulli functionals. The Hadamard gradient is introduced for the formulation of the gradient flow system, and the concept of Hessian is introduced for analyzing the solvability of this system. The section summarizes the main results of the paper, including expressions for the Hessians of both functionals. We also describe the application of the Nash–Moser method to establish the solvability of the gradient flow equations.

Plotnikov P.I. , Sokolowski J.
Geometric framework for gradient flow in shape optimization
Успехи математических наук. 2026. Т.81. №3(489). С.51-150. DOI: 10.4213/rm10296 OpenAlex

Submitted:	Dec 8, 2025
Published print:	May 31, 2026

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