2020/11/08 by Michael Frazier, Frazier, Michael W., Igor E. Verbitsky +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2011.04083
We give bilateral pointwise estimates for positive solutions of the equation\n\
left
beginaligned -
triangle u amp; =
omega u
,
,amp; amp;\n
mboxin
,
,
Omega,
quad u
ge 0,
u amp; = f
,
, amp; amp;
mboxon
,
,\n
partial
Omega ,
endaligned
right. in a bounded uniform\ndomain \Ω\⊂ bf Rn, where \ω is a locally finite Borel\nmeasure in \Ω, and f\≥ 0 is integrable with respect to harmonic\nmeasure d Hx on \∂\Ω.\n We also give sufficient and matching necessary conditions for the existence\nof a positive solution in terms of the exponential integrability of M* (m\n\ω)(z)=\∫_\Ω M(x, z) m(x) , d \ω (x) on \∂\Ω with\nrespect to f , d Hx0, where M(x, \⋅) is Martin's function with pole\nat x0\∈ \Ω, m(x)=\min (1, G(x, x0)), and G is Green's function.\n These results give bilateral bounds for the harmonic measure associated with\nthe Schr "odinger operator - triangle - \ω on \Ω, and in the\ncase f=1, a criterion for the existence of the gauge function. Applications\nto elliptic equations of Riccati type with quadratic growth in the gradient are\ngiven.\n