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Regularity of the Optimal Sets for a Class of Integral Shape Functionals.
Giuseppe Buttazzo1, Francesco Paolo Maiale2, Dario Mazzoleni3
1Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, 56127 Pisa, Italy.
Archive for Rational Mechanics and Analysis
|September 9, 2024
Summary
This study establishes the first regularity theorem for free boundaries in shape optimization, proving Lipschitz continuity and non-degeneracy for optimal domains. It introduces a new stability notion for the one-phase problem, enabling dimension reduction and proving C^infinity regularity for smooth data.
Area of Science:
- Shape Optimization
- Partial Differential Equations
- Free Boundary Problems
Background:
- Shape optimization problems often involve integral functionals dependent on solutions to partial differential equations (PDEs).
- Minimality conditions for domains in these problems do not always translate to variational problems for a single state function.
- The regularity of free boundaries in such problems is a critical aspect for understanding solution behavior.
Purpose of the Study:
- To establish the first regularity theorem for the free boundary of solutions in shape optimization problems with integral functionals.
- To analyze the behavior of solutions and their free boundaries under specific conditions, including affine cost functions.
- To develop new methods for estimating the dimension of singular sets and proving regularity of the free boundary.
Main Methods:
- Focusing on affine cost functions and solutions to PDEs with Dirichlet boundary conditions.
- Utilizing inwards/outwards optimality to establish Lipschitz continuity and non-degeneracy of the optimal state function.
- Employing stability with respect to smooth vector fields, triple blow-up analysis, and a new formulation of stability for the one-phase problem.
- Combining a higher-order Boundary Harnack principle and a viscosity approach for regularity results.
Main Results:
- Proved the first regularity theorem for the free boundary in shape optimization problems involving integral functionals.
- Established Lipschitz continuity and non-degeneracy of the optimal state function `u`.
- Demonstrated blow-up sequences converging to homogeneous stable solutions of the one-phase Bernoulli problem.
- Decomposed the domain into singular and regular parts based on blow-up limits.
- Developed a dimension reduction principle by introducing a new stability notion for the one-phase problem.
- Proved C^infinity regularity of the regular part of the free boundary for smooth data.
Conclusions:
- The study provides fundamental regularity results for free boundaries in a class of shape optimization problems.
- The developed techniques, including a new stability notion and dimension reduction principle, offer powerful tools for analyzing free boundary problems.
- The findings contribute to a deeper understanding of the geometric properties of optimal shapes and their associated solutions.
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