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Explore the asymptotic behavior of sphere-valued Sobolev maps in this 57-minute lecture from the Erwin Schrödinger International Institute for Mathematics and Physics. Examine how p-energy approaches the critical Sobolev exponent, which corresponds to the codimension of the singular set of these mathematical objects. Learn about recent research demonstrating compactness and Gamma-convergence of renormalized p-energy to the area functional of appropriate dimension. Discover how this work recovers classical results by Hardt and Lin regarding the convergence of energy densities for p-energy minimizing maps with fixed boundary conditions as p approaches the critical exponent. Understand how these findings establish the p-energy analog of the celebrated Ginzburg-Landau energy work by Alberti, Baldo, and Orlandi, extending the results to general dimension and codimension. This advanced mathematical presentation was delivered as part of the Thematic Programme on Free Boundary Problems and represents collaborative research with Mattia Freguglia and Nicola Picenni.
