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Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators. (English) Zbl 1028.31003
Three loosely connected questions for second order strictly elliptic differential operators \(\mathcal L\) (on a Riemannian manifold) are treated in this paper. The first result answers a question of Y. Pinchover [Math. Ann. 314, 555-590 (1999; Zbl 0928.35010)]: Let \(M\) be a connected, non-compact (but not necessarily complete) Riemannian manifold \(M\) and \(\mathcal L\) be a second-order elliptic differential operator which is symmetric w.r.t. a smooth density (or, more generally, which is in a certain sense quasi-symmetric). Then there exists a positive \(\mathcal L\)-harmonic function \(u\) such that for the Green function \(G(x,y)/u(x) x @>>x\to\partial_M>0\) (\(\partial_M\) denotes the point at infinity). In this result the (quasi-)symmetry is essential. For \(\mathbb R^n\), \(n\geq 3\), a smooth vector field \(V\) is constructed such that for \(\mathcal L := \Delta + V\cdot\nabla\) the pair \((\mathbb R^n,\mathcal L)\) is transient, all \(\mathcal L\)-harmonic functions are constant but the Green function of \(\mathcal L\) (with a pole at \(p\in M\)) does not vanish at infinity.
The second result concerns critical points of certain harmonic functions related to the operator \(\mathcal L := \Delta + \nabla\phi\cdot\nabla\) where \(\phi\) is \(\mathbb Z^n\)-periodic. It is shown that \(\mathcal L F = 0\) has for \(n=1,2\) always a solution \(F\) which is a diffeomorphism. For \(n\geq 3\) a concrete positive periodic \(C^\infty\)-function \(\phi\) is constructed s.t. \(\mathcal L F = 0\) exists but has critical points.
The last two sections are devoted to stability properties of the Green functions for second order elliptic operators of the form \[ \mathcal L_ju = \text{div}(A_j(\nabla u)) - B_j\cdot\nabla u - V_j\cdot u,\quad (j=1,2), \] where \(A_j\) are bounded Borel and uniformly accretive sections of \(\text{End}(T(M))\), \(B_j\) are \(V_0\)-bounded Borel vector fields and \(V_j\) are \(V_0\)-bounded Borel functions. Under certain conditions on these coefficients it is shown that the Green functions of the operators \(\mathcal L_j\) are mutually comparable. The proof is essentially based on the methods developed by the author in [J. Anal. Math. 72, 45-92 (1997; Zbl 0944.58016)].

31C12 Potential theory on Riemannian manifolds and other spaces
31C35 Martin boundary theory
58J05 Elliptic equations on manifolds, general theory
60J45 Probabilistic potential theory
60J60 Diffusion processes
35J25 Boundary value problems for second-order elliptic equations
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