TY - JOUR

T1 - Mappings of Least Dirichlet Energy and their Hopf Differentials

AU - Iwaniec, Tadeusz

AU - Onninen, Jani

N1 - Funding Information:
Iwaniec was supported by the NSF grant DMS-0800416 and the Academy of Finland project 1128331. Onninen was supported by the NSF grant DMS-1001620.

PY - 2013/8

Y1 - 2013/8

N2 - The paper is concerned with mappings h: X onto→ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X) of the energy-minimal mappings is established within the H̄2(X, Y) of strong limits of homeomorphisms in the Sobolev space W1,2(X, Y), a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in X leads to the Hopf differential hzhz̄dz⊗dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle: A mapping h ∈ H̄2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in X (cognate with cracks). Slits are triggered by points of concavity of Y. They originate from ∂X and advance along vertical trajectories of the Hopf differential toward X where they eventually terminate, so no crosscuts are created.

AB - The paper is concerned with mappings h: X onto→ between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X) of the energy-minimal mappings is established within the H̄2(X, Y) of strong limits of homeomorphisms in the Sobolev space W1,2(X, Y), a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in X leads to the Hopf differential hzhz̄dz⊗dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle: A mapping h ∈ H̄2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in X (cognate with cracks). Slits are triggered by points of concavity of Y. They originate from ∂X and advance along vertical trajectories of the Hopf differential toward X where they eventually terminate, so no crosscuts are created.

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U2 - 10.1007/s00205-012-0606-4

DO - 10.1007/s00205-012-0606-4

M3 - Article

AN - SCOPUS:84878199245

SN - 0003-9527

VL - 209

SP - 401

EP - 453

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -