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The p-harmonic transform beyond its natural domain of definition. (English) Zbl 1081.35048

Summary: The p-harmonic transforms are the most natural nonlinear counterparts of the Riesz transforms in n . They originate from the study of the p-harmonic type equation

div|u| p-2 u=div𝔣,

where 𝔣:Ω n is a given vector field in 𝔏 q (Ω, n ) and u is an unknown function of Sobolev class 𝒲 0 1,p (Ω, n ), p+q=pq. The p-harmonic transform p : p (Ω, n ) assigns to 𝔣 the gradient of the solution: p 𝔣=u p (Ω, n ). More general PDE’s and the corresponding nonlinear operators are also considered. We investigate the extension and continuity properties of the p-harmonic transform beyond its natural domain of definition. In particular, we identify the exponents λ>1 for which the operator p : λq (Ω, n ) λp (Ω, n ) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution u λp (Ω, n ) fails when λ exceeds certain critical value. In case p=n=dimΩ, there is no uniqueness in 𝒲 1,λn ( n ) for any λ>1.


MSC:
35J99Elliptic equations and systems
44A15Special transforms (Legendre, Hilbert, etc.)
47H99Nonlinear operators