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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 33, Fasc. 1,
pages 29 - 44
 

COMPARISON OF HARMONIC KERNELS ASSOCIATED WITH A CLASS OF SEMILINEAR ELLIPTIC EQUATIONS

Mahmoud Ben Fredj
Khalifa El Mabrouk

Abstract: Let D be a smooth domain in ℝN , N ≥ 3, and let f be a positive continuous function on ∂D . Under some assumptions on φ , it is shown that the problem Δu = 2φ(u) in D and u = f on ∂D admits a unique solution which will be denoted by H φf D . Given two functions φ and ψ , our main goal in this paper is to investigate the existence of a constant c > 0 such that

1 φ ψ φ cH Df ≤ HDf ≤ cH Df.

2000 AMS Mathematics Subject Classification: Primary: 60J45, 60H30; Secondary: 60J35, 35J25.

Keywords and phrases: Brownian motion, Green operator, Dirichlet problem, harmonic kernel, semilinear problem.

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