Abstract.Ventcel's boundary condition is the most general lateral condition determing Markov extension of a minimal diffusion process governed by a diffusion operator on a smooth Euclidean or Riemannian domain. The boundary operator is given as a linear combination of a reflection operator, the boundary trace of the interior operator and a certain second order (possibly degenerate) elliptic type integro-differential operator. A Ventcel'-Višik boundary condition is local one without term of the boundary trace of the interior operator. Since the boundary operator has a term of second order differential operator, it is hard to apply a modern approach based on function spaces such as Sobolev space or weighted Hölder space to construct a weak solution or a fundamental solution. Using the classical method based on solving Volterra type integral equations with singular kernel, we construct a fundamental solution to diffusion equations with Ventcel'-Višik boundary condition under low regularity of coefficients.

Received: April 20, 2005

AMS Subject Classification: 35K20, 45D05

Key Words and Phrases: diffusion equation, Ventcel'-Višik boundary condition, fundamental solution

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 2