RadóKneserChoquet Theorem for simply connected domains (pharmonic setting)
Iwaniec, T., & Onninen, J. (2019). RadóKneserChoquet Theorem for simply connected domains (pharmonic setting). Transactions of the American Mathematical Society, 371(4), 23072341. https://doi.org/10.1090/tran/7348
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© 2018 American Mathematical Society
A remarkable result known as Rad´oKneserChoquet
theorem asserts that the harmonic extension of a homeomorphism
of the boundary of a Jordan domain ⌦ ⇢ R2 onto the boundary
of a convex domain Q ⇢ R2 takes ⌦ di↵eomorphically onto Q .
Numerous extensions of this result for linear and nonlinear elliptic
PDEs are known, but only when ⌦ 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 to extend
this theorem to the setting on 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 di↵erent than that of Jordan domains.
One must circumvent the inherent topological diculties arising
near the boundary.
Our main Theorem 4 is the key to establishing approximation
of monotone Sobolev mappings with di↵eomorphisms. This, in
turn, leads to the existence of energyminimal deformations in the
theory of Nonlinear Elasticity. Hence the usefulness of Theorem
4. We do not enter these applications here, but refer the reader to
Section 1.2, for comments. .
...
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