On \(p\)-harmonic measures in half-spaces (Q2315236)

A direct generalization of the classical harmonic measure to a class of nonlinear equations, including the \(p\)-Laplace equation \[ \nabla\cdot(|\nabla\omega|^{p-2}\nabla\omega)= 0, \] was introduced by [\textit{S. Granlund} et al., Ann. Acad. Sci. Fenn., Ser. A I, Math. 7, 233--247 (1982; Zbl 0468.30015)]. Given a domain \(\Omega\) in \(\mathbb{R}^N\) and a set \(C\) on the boundary \(\partial\Omega\), the \(p\)-harmonic measure of \(C\) in \(\Omega\) is defined as the (upper Perron) solution \(\omega_p(x)= \omega_p(x;C,\Omega)\) of the problem \[ \begin{cases} \nabla\cdot(|\nabla\omega_p|^{p-2} \nabla\omega_)=0\quad & \text{in }\Omega,\\ \omega_p= 1\quad & \text{in }C,\\ \omega_p= 0\quad & \text{on }\partial\Omega\setminus C.\end{cases} \] For \(p=2\) we have the Laplace equation \(\Delta\omega_2=0\) and the classical harmonic measure. Always \(0\le\omega_p\le 1\), but for \(p\ne 2\) it is not a measure. By [\textit{J. J. Manfredi} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 4, No. 2, 357--373 (2005; Zbl 1105.31002)], \(\omega_p\) is not subadditive even at the zero level! The authors consider the situation when \(\Omega\) is the upper half-space \(\mathbb{R}^N_+= \{(x_1,\dots, x_N)\mid x_N>0\}\) \(\Omega\) of \(\mathbb{R}^N\) and \(C\) is the \(N-1\)-dimensional ball \(B_\delta= \{(x_1,\dots, x_{N-1},0)\mid x^2_1+\cdots x_{N-1}^2<\delta^2\}\) of radius \(\delta\), \(0<\delta\le 1\). Now \(N\ge 2\) and \(1<p<\infty\). The remarkable estimates \[ C_1 \delta^{\alpha(p,N)}\le \omega_p(x_0; B_\delta,\mathbb{R}^N_+)\le C_2 \delta^{\alpha(p,N)} \] are obtained. Here \(x_0= (0,\dots, 0,1)\) and the constants depend only on \(p\) and \(N\) \((0<C_1<C_2<\infty)\). Notice that the exponent \(\alpha(p,N)\) is the same in the lower and upper estimates. It is difficult to determine \(\alpha(p,N)\) when \(p\ne 2\). The plane case \(N=2\) is explicitly known, see [\textit{N. L. P. Lundström} and \textit{J. Vasilis}, Ann. Acad. Sci. Fenn., Math. 38, No. 1, 351--366 (2013; Zbl 1271.31002)]. In the conformally invariant case \(p=N\), we have \(\alpha(N,N)= 1\). The corresponding \(\omega_N\) is known, see [the reviewer, J. Differ. Equations 58, 307--317 (1985; Zbl 0534.34042)]. For general \(p\), the authors succeed in improving previously known estimates of \(\alpha(p,N)\). The main tool of the present work is a so-called quasi-radial solution \(r^kf(\theta)\), where \(\theta\) is the ``co-latitude'' angle in spherical coordinates. Such functions have been used by \textit{I. N. Krol'} and \textit{V. G. Maz'ya} [Trans. Mosc. Math. Soc. 26, 73--93 (1974; Zbl 0281.35013)], Tolksdorf, Aronsson, Veron, and others. A new feature of the present paper is the use of a simple, but ingeneous, shooting method, replacing complicated tools like boundary Harnack inequalities. Thus the problem is reduced to an ordinary differential equation.