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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 \(\mathbb{R}^n\). They originate from the study of the \(p\)-harmonic type equation \[ \text{div} |\nabla u|^{p-2}\nabla u=\operatorname{div}{\mathfrak f}, \] where \({\mathfrak f}: \Omega\to\mathbb{R}^n\) is a given vector field in \({\mathfrak L}^q(\Omega, \mathbb{R}^n)\) and \(u\) is an unknown function of Sobolev class \({\mathcal W}_0^{1,p}(\Omega,\mathbb{R}^n)\), \(p+q=pq\). The \(p\)-harmonic transform \({\mathcal H}_p: {\mathcal L}^p(\Omega,\mathbb{R}^n)\) assigns to \({\mathfrak f}\) the gradient of the solution: \({\mathcal H}_p{\mathfrak f}=\nabla u\in{\mathcal L}^p(\Omega,\mathbb{R}^n)\). 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 \(\lambda> 1\) for which the operator \({\mathcal H}_p:{\mathcal L}^{\lambda q} (\Omega,\mathbb{R}^n)\to {\mathcal L}^{\lambda p}(\Omega,\mathbb{R}^n)\) is well defined and remains continuous. Rather surprisingly, the uniqueness of the solution \(\nabla u\in{\mathcal L}^{\lambda p}(\Omega,\mathbb{R}^n)\) fails when \(\lambda\) exceeds certain critical value. In case \(p=n=\dim \Omega\), there is no uniqueness in \({\mathcal W}^{1,\lambda n}(\mathbb{R}^n)\) for any \(\lambda>1\).

MSC:
35J99 Elliptic equations and elliptic systems
44A15 Special integral transforms (Legendre, Hilbert, etc.)
47H99 Nonlinear operators and their properties
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