Luigi D'Onofrio (Universita' degli Studi di Napoli)
The p-harmonic transform beyond its natural domain of
definition, interpolation and continuity
Abstract.
The p-harmonic transforms are the most natural nonlinear counterparts of
the
Riesz transforms. They originate from the study of the p-harmonic type
equation
where the right hand side is the divergence of a vector field that is
integrable with power q (p+q=pq p,q>1). The p-harmonic transform
assigns to f the gradient of the solution of the p-harmonic type equation:
H(f)=D u. 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 (f is
integrable
with power lambda q and the gradient of the solution is integrable with
power
lambda p). Rather surprisingly, the uniqueness of the solution fails when
lambda exceeds certain critical value. In case p=n=dimension of the domain
we
are considering, there is no uniqueness for any lambda>1.
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