### Abstract

A remarkable result known as the Radó-Kneser-Choquet theorem asserts that the harmonic extension of a homeomorphism of the boundary of a Jordan domain Ω ⊂ R2 onto the boundary of a convex domain Q ⊂ R2 takes Ω diffeomorphically onto Q. Numerous extensions of this result for linear and nonlinear elliptic PDEs are known, but only when Ω is a Jordan domain or, if not, under additional assumptions on the boundary map. On the other hand, the newly developed theory of Sobolev mappings between Euclidean domains and Riemannian manifolds demands extending this theorem to the setting of simply connected domains. This is the primary goal of our article. The class of the p-harmonic equations is wide enough to satisfy those demands. Thus we confine ourselves to considering the p-harmonic mappings. The situation is quite different from that of Jordan domains. One must circumvent the inherent topological difficulties arising near the boundary. Our main theorem is the key to establishing approximation of monotone Sobolev mappings with diffeomorphisms. This, in turn, leads to the existence of energy-minimal deformations in the theory of nonlinear elasticity.

Original language | English (US) |
---|---|

Pages (from-to) | 2307-2341 |

Number of pages | 35 |

Journal | Transactions of the American Mathematical Society |

Volume | 371 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Harmonic mappings
- Monotone mappings
- P-harmonic equation

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**RadÓ-kneser-choquet theorem for simply connected domains (P-harmonic setting).** / Iwaniec, Tadeusz; Onninen, Jani Kristian.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 371, no. 4, pp. 2307-2341. https://doi.org/10.1090/tran/7348

}

TY - JOUR

T1 - RadÓ-kneser-choquet theorem for simply connected domains (P-harmonic setting)

AU - Iwaniec, Tadeusz

AU - Onninen, Jani Kristian

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A remarkable result known as the Radó-Kneser-Choquet theorem asserts that the harmonic extension of a homeomorphism of the boundary of a Jordan domain Ω ⊂ R2 onto the boundary of a convex domain Q ⊂ R2 takes Ω diffeomorphically onto Q. Numerous extensions of this result for linear and nonlinear elliptic PDEs are known, but only when Ω is a Jordan domain or, if not, under additional assumptions on the boundary map. On the other hand, the newly developed theory of Sobolev mappings between Euclidean domains and Riemannian manifolds demands extending this theorem to the setting of simply connected domains. This is the primary goal of our article. The class of the p-harmonic equations is wide enough to satisfy those demands. Thus we confine ourselves to considering the p-harmonic mappings. The situation is quite different from that of Jordan domains. One must circumvent the inherent topological difficulties arising near the boundary. Our main theorem is the key to establishing approximation of monotone Sobolev mappings with diffeomorphisms. This, in turn, leads to the existence of energy-minimal deformations in the theory of nonlinear elasticity.

AB - A remarkable result known as the Radó-Kneser-Choquet theorem asserts that the harmonic extension of a homeomorphism of the boundary of a Jordan domain Ω ⊂ R2 onto the boundary of a convex domain Q ⊂ R2 takes Ω diffeomorphically onto Q. Numerous extensions of this result for linear and nonlinear elliptic PDEs are known, but only when Ω is a Jordan domain or, if not, under additional assumptions on the boundary map. On the other hand, the newly developed theory of Sobolev mappings between Euclidean domains and Riemannian manifolds demands extending this theorem to the setting of simply connected domains. This is the primary goal of our article. The class of the p-harmonic equations is wide enough to satisfy those demands. Thus we confine ourselves to considering the p-harmonic mappings. The situation is quite different from that of Jordan domains. One must circumvent the inherent topological difficulties arising near the boundary. Our main theorem is the key to establishing approximation of monotone Sobolev mappings with diffeomorphisms. This, in turn, leads to the existence of energy-minimal deformations in the theory of nonlinear elasticity.

KW - Harmonic mappings

KW - Monotone mappings

KW - P-harmonic equation

UR - http://www.scopus.com/inward/record.url?scp=85062176759&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062176759&partnerID=8YFLogxK

U2 - 10.1090/tran/7348

DO - 10.1090/tran/7348

M3 - Article

AN - SCOPUS:85062176759

VL - 371

SP - 2307

EP - 2341

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -