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Continuous
symmetrizations and uniqueness of solutions to nonlocal
equations
Matías G. Delgadino and Mary Vaughan |
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Vol. 18 (2025), No. 10, 2325–2366
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Abstract |
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We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long-time dynamics of a higher-order nonlocal equation which models the growth of symmetric cracks in an elastic medium. |
Keywords
continuous Steiner symmetrizations, nonlocal seminorms,
fractional thin-film equation, higher-order equations
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Mathematical Subject Classification
Primary: 35C06, 35G20, 35R11
Secondary: 35B40, 74G30
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Milestones
Received: 14 December 2023
Revised: 18 September 2024
Accepted: 15 November 2024
Published: 4 November 2025
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Authors |
| Matías G. Delgadino | |
| Department of Mathematics University of Texas at Austin Austin, TX United States |
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| Mary Vaughan | |
| Department of Mathematics Texas State University San Marcos United States |
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| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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