In GFT we seek, as a generalization of the Riemann Mapping Problem, homeomorphisms that minimize certain energy integrals. No boundary values of such homeomorphisms are prescribed. This is interpreted as saying that the deformations are allowed to slip along the boundary, known as traction free problems. This leads us to determine the infimum of a given energy functional among homeomorphisms from X onto Y. Even in the basic case of the Dirichlet energy, it is certainly unrealistic to require that the infimum energy be attained within the class of homeomorphisms. Of course, enlarging the set of the admissible mappings can change the nature of the energy-minimal solutions. To avoid the Lavrentiev phenomenon one is forced to show that the minimizing sequence actually converges strongly which is the subject of my talk.
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