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Lower estimates for a perturbed Green function. (English) Zbl 1158.35022

In this paper, which is quite technical, the authors prove the existence, and estimates for lower bounds, of a perturbed Green function on a locally connected, locally compact space \(X\), which satisfies other conditions as well.
If \(g(x,y)\) is the Green function for the Laplacian, \(\Delta \), on \( \mathbb{R}^{n}\), and if \(\mu \) is a measure on \(\mathbb{R}^{n}\) such that the map \(x\mapsto \int_{\mathbb{A}}g(x,y)d\mu (y)\) is finite and continuous for all bounded open sets \(A\subset \mathbb{R}^{n}\), then the equation of the perturbed Laplacian, \(\Delta u-u\mu =0\), for \(n>2\), also has a positive Green function denoted by \(^{\mu }g(x,y)\).
To introduce their main result, the authors first present a special case: for \(^{\mu }g(x,y)\) as above, and for \(R\geq 3\left\| x-y\right\| >0\), there is a constant \(C(n)>0\), so that \[ \frac{^{\mu }g(x,y)}{g(x,y)}\geq \exp \left\{ -C(n)\left( 1+\int_{B(x,R)}g(x,z)d\mu (z)+\int_{B(y,R)}g(y,z)d\mu (z)\right) \right\} . \] They discuss consequences for a measure of the form \(d\mu (x)=V(x)dx\), with \(V(x)\leq \frac{V_{0}}{1+\left| x\right| ^{\gamma }}\), and also previous results of their own and of many other mathematicians including M. Murata [Duke Math J. 53, 869–943 (1986; Zbl 0624.35023)], Q. S. Zhang and Z. Zhao [Ill. J. Math. 44, No. 3, 556–573 (2000; Zbl 0985.35016)], Y. Pinchover [Differ. Integral Equ. 5, 481–493 (1992; Zbl 0772.35015)], and (in a later section) V. Liskevich and Y. Semenov [J. Lond. Math. Soc., II. Ser. 62, No. 2, 521–543 (2000; Zbl 1020.35026)].
In section 2 of the paper, definitions, including the exact spaces involved, are given; it is shown that the perturbed Green function exists on the space of central interest, a \(\mathcal{P}\)-harmonic Bauer space. In section 3 a preliminary estimate on \(\frac{^{\mu }g(x,y)}{g(x,y)}\) establishes upper and lower bounds, this estimate is then used to prove the main theorem in section 4. The last section of the paper deals with applications of the main theorem.

MSC:

35J15 Second-order elliptic equations
35A08 Fundamental solutions to PDEs
35B35 Stability in context of PDEs
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