The Nitsche phenomenon for weighted Dirichlet energy

The present paper arose from recent studies of energy-minimal deformations of planar domains. We are concerned with the Dirichlet energy. In general the minimal mappings need not be homeomorphisms. In fact, a part of the domain near its boundary may collapse into the boundary of the target domain. In mathematical models of nonlinear elasticity this is interpreted as interpenetration of matter. We call such occurrence the Nitsche phenomenon, after Nitsche’s remarkable conjecture (now a theorem) about existence of harmonic homeomorphisms between annuli. Indeed the round annuli proved to be perfect choices to grasp the nuances of the problem. Several papers are devoted to a study of deformations of annuli. Because of rotational symmetry it seems likely that the Dirichlet energy-minimal deformations are radial maps. That is why we confine ourselves to radial minimal mappings. The novelty lies in the presence of a weight in the Dirichlet integral. We observe the Nitsche phenomenon in this case as well, see our main results Theorem 1.4 and Theorem 1.7. However, the arguments require further considerations and new ingredients. One must overcome the inherent difficulties arising from discontinuity of the weight. The Lagrange–Euler equation is unavailable, because the outer variation violates the principle of none interpenetration of matter. Inner variation, on the other hand, leads to an equation that involves the derivative of the weight. But our weight function is only measurable which is the main challenge of the present paper.

The subelliptic ∞-Laplace system on Carnot–Carathéodory spaces

Given a Carnot–Carathéodory space with associated frame of vector fields , we derive the subelliptic ∞-Laplace system for mappings , which reads in the limit of the subelliptic p-Laplacian as . Here is the horizontal gradient and is the projection on its nullspace. Next, we identify the variational principle characterizing the subelliptic ∞-Laplacian system, which is the “Euler–Lagrange PDE” of the supremal functional for an appropriately defined notion of horizontally ∞-minimal mappings. We also establish a maximum principle for for solutions to the subelliptic ∞-Laplacian system. These results extend previous work of the author [J. Differential Equations 253 (2012), no. 7, 2123–2139; Proc. Amer. Math. Soc., to appear] on vector-valued calculus of variations in L ∞ from the Euclidean to the subelliptic setting.