Energy Minimisers with Prescribed Jacobian.

Area of Science:

• Complex Analysis
• Geometric Function Theory
• Calculus of Variations

Background:

• Investigating properties of planar maps is crucial in various mathematical fields.
• Understanding energy minimization problems provides insights into the behavior of mappings.

Purpose of the Study:

• To analyze planar maps with a fixed radially symmetric Jacobian that minimize the 2p-Dirichlet energy.
• To determine how a specific quantity influences the symmetry, uniqueness, and regularity of these energy minimizers.

Main Methods:

• Minimization of the 2p-Dirichlet energy for planar maps.
• Analysis of the Jacobian condition and its implications on map properties.
• Study of the impact of a controlling quantity on map characteristics.

Main Results:

• A quantity is identified that governs the symmetry and uniqueness of minimizers.
• If the quantity is small, minimizers are symmetric and unique.
• If the quantity is large, multiple non-symmetric minimizers may exist with optimal regularity.
• Infinite quantity leads to generically lower regularity minimizers.

Conclusions:

• The study provides a comprehensive understanding of minimizers for planar maps with prescribed Jacobian.
• Results offer a negative answer to a question posed by Hélein regarding map regularity.
• Findings extend to more general domains and boundary conditions.