The Perron method for \(p\)-harmonic functions in unbounded sets in \(\mathbf {R}^n\) and metric spaces
The authors extend the classical notions of \(p\)-subharmonic and \(p\)-superharmonic function, that is a function satisfying, respectively, the inequalities \(-\Delta_p u\leq 0\) and \(-\Delta_p u\geq 0\) in the Euclidean space \(\mathbb{R}^n\), to the setting of a metric space \((X,\mathcal{M},\mu,d)\), equipped with a metric \(d\) and with a complete measure \(\mu\) defined on the completion \(\mathcal{M}\) of the Borel \(\sigma\)-algebra. To this end, the authors introduce the notion of \(p\)-weak upper subgradient and the related notions of Newtonian and Dirichlet spaces on \(X\). Then, given an open set \(\Omega\) in \(X\) which satisfies a \(p\)-parabolic condition if unbounded, the authors apply the Perron method in this setting to find \(p\)-harmonic functions on \(\Omega\) satisfying, in the sense specified below, an assigned boundary condition on \(\partial \Omega\). Going into details, given a real-extended function \(f\) defined on \(\partial \Omega\), the authors consider the set \(\mathcal{U}_f(\Omega)\) of all bounded below superharmonic functions \(u\) in \(\Omega\) such that \(\liminf_{y\rightarrow x}u(y)\geq f(x)\), for all \(x\in \partial \Omega\), and the set \(\mathcal{L}_f(\Omega)\) of all bounded above subharmonic functions \(u\) in \(\Omega\) such that \(\limsup_{y\rightarrow x}u(y)\leq f(x)\), for all \(x\in \partial \Omega\). Then the authors find sufficient conditions on \(f\) in order to \[ P_\Omega(x):=\inf_{u\in \mathcal{U}_f(\Omega)}u(x)=\sup_{u\in \mathcal{L}_f(\Omega)}u(x)\in \mathbb{R}, \] for all \(x\in \Omega\). When the above equality holds, the function \(f\) is said to be resolutive with respect to \(\Omega\), and \(P_\Omega\) is called the Perron solution of \(f\). After introducing a new suitable concept of \(p\)-capacity for the subsets of \(\overline{\Omega}\), the authors prove that a function \(f:\overline{\Omega}\rightarrow \overline{\mathbb{R}}\) is resolutive with respect to \(\Omega\) if the following conditions hold: {\parindent=7mm\begin{itemize}\item[1)] \ \(f_{\mid\Omega}\) has a \(p\)-summable upper gradient; \item[2)] \ for each \(\varepsilon>0\), there is a relatively open set \(U\) in \(\overline{\Omega}\setminus \{\infty\}\), with \(p\)-capacity less than \(\varepsilon\), such that the restriction of \(f\) to \((\overline{\Omega}\setminus \{\infty\})\setminus U\) is continuous and real valued. \end{itemize}} It is also proved that the Perron solution of \(f\) coincides with the \(p\)-harmonic extension of \(f\) to \(\Omega\), which is defined as the continuous solution of a suitable obstacle problem. In addition, the authors prove that continuous functions defined on the boundary of \(\Omega\) are also resolutive with respect to \(\Omega\) and that the Perron solution of \(f\) is invariant under perturbations of \(f\) on subsets of \(\partial \Omega\setminus \{\infty\}\) of \(p\)-capacity zero.
scientific article; zbMATH DE number 6851617
scientific article