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Gauge functions, eikonal equations and Bôcher’s theorem on stratified Lie groups. (English) Zbl 1137.31003
Let \(\mathcal{L}=\sum_1^mX_i^2\) be a sub-Laplacian on a Carnot group \(G\) (i.e., \(G=\mathbb{R}^N\) equipped with a family of dilations \(\{\delta_\lambda\}_{\lambda>0}\) of the form \(\delta_\lambda(x_1,\ldots,x_N)=(\lambda^{\alpha_1}x_1,\ldots,\lambda^{\alpha_N}x_N),\) with integers \(1=\alpha_1\leq\ldots\leq\alpha_N\)) and let \(\nabla_{\mathcal{L}}=(X_1,\ldots,X_m)\) be the intrinsic gradient related to \(\mathcal{L}.\) Let \(d\) be a homogeneous norm on \(G.\) A homogeneous norm \(d\) is said to be \(\mathcal{L}\)-gauge if there exists a constant \(\gamma\not=0\) such that \(d^\gamma\) is \(\mathcal{L}\)-harmonic in \(G\setminus\{0\}.\)
The first main result (Theorem 1.1) says that if \(d\) is a \(\mathcal{L}\)-gauge satisfying the \(\mathcal{L}\)-eikonal equation \(| \nabla_{\mathcal{L}}d| =1\) on \(G\setminus\{0\}\) then \(G\) is the Euclidean group \((\mathbb{R}^N,+).\)
Let \(d\) be a \(\mathcal{L}\)-gauge and define the kernels: \(K=| \nabla_{\mathcal{L}}d| ^2\) and \(\mathcal{K}=K/| \nabla d| ,\) where \(\nabla\) is the standard gradient vector. Put \(m_r(u)=\int_{\partial B_d(0,r)}u\mathcal{K}\,dH^{N-1},\) where \(H^{N-1}\) is the \((N-1)\)-Hausdorff measure in \(\mathbb{R}^N\) and \(B_d(x,r)\) is the \(d\)-ball of center \(x\) and radius \(r,\) \(B_d(x,r)=\{y\in G:d(x^{-1}y)<r\}.\) With this notation the authors prove the following result (Theorem 1.5): Let \(d\) be a \(\mathcal{L}\)-gauge and let \(\gamma\) be the related exponent. If \(u\) is \(\mathcal{L}\)-harmonic in the unit punctured ball \(B_d(0,1)\setminus\{0\},\) then there exist real constants \(a,b\) such that \(r^{\gamma-1}m_r(u)=ar^\gamma+b\) for every \(r\in(0,1).\)
As a corollary, if \(d\) is a \(\mathcal{L}\)-gauge with exponent \(\gamma,\) then the function \(\Gamma=\beta_dd^{2-Q},\) where \(Q\) is a homogeneous dimension of \(G,\) is the fundamental solution (with pole at \(0\)) for \(\mathcal{L}.\) The constant \(\beta_d=1/(Q(Q-2)\int_{B_d(0,1)}K).\)
The next main result is Theorem 1.8: Let \(d\) be a \(\mathcal{L}\)-gauge and let \(w\) be a nonnegative \(\mathcal{L}\)-harmonic function in \(B_d(0,1)\setminus\{0\}.\) Assume that \(w\) is continuous up to the boundary of \(B_d(0,1)\) and \(w(x)=0\) for every \(x\in\partial B_d(0,1).\) Then \(w\) is radially symmetric. More precisely, there is a real constant \(a>0\) such that \(w(x)=a(d^{2-Q}(x)-1)\) for every \(x\in B_d(0,1)\setminus\{0\}.\)
As a corollary, from the above theorem the authors obtain, using the left-invariance of \(\mathcal{L},\) the Bôcher-type theorem for nonnegative \(\mathcal{L}\)-harmonic functions in the punctured open sets \(\Omega\setminus\{x_0\}\subset G,\) where \(x_0\in\Omega\) is fixed.

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
35J70 Degenerate elliptic equations
35H20 Subelliptic equations
Full Text: DOI
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