On Second Order Elliptic Equations with a Small Parameter

Mark Freidlin; Wenqing Hu

doi:10.1080/03605302.2013.812658

On Second Order Elliptic Equations with a Small Parameter

Mark Freidlin, Wenqing Hu

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

The Neumann problem with a small parameter (Formula presented.) is considered in this paper. The operators L₀and L₁are self-adjoint second order operators. We assume that L₀has a non-negative characteristic form and L₁is strictly elliptic. The reflection is with respect to inward co-normal unit vector γ^ε(x). The behavior of lim_ε↓0u^ε(x) is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.

Original language	English (US)
Pages (from-to)	1712-1736
Number of pages	25
Journal	Communications in Partial Differential Equations
Volume	38
Issue number	10
DOIs	https://doi.org/10.1080/03605302.2013.812658
State	Published - Oct 2013

Keywords

Averaging principle
Diffusion processes on a graph
Equations with non-negative characteristic form
Second order equations with a small parameter

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10.1080/03605302.2013.812658

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abstract = "The Neumann problem with a small parameter (Formula presented.) is considered in this paper. The operators L0and L1are self-adjoint second order operators. We assume that L0has a non-negative characteristic form and L1is strictly elliptic. The reflection is with respect to inward co-normal unit vector γε(x). The behavior of limε↓0uε(x) is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.",

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AU - Freidlin, Mark

AU - Hu, Wenqing

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N2 - The Neumann problem with a small parameter (Formula presented.) is considered in this paper. The operators L0and L1are self-adjoint second order operators. We assume that L0has a non-negative characteristic form and L1is strictly elliptic. The reflection is with respect to inward co-normal unit vector γε(x). The behavior of limε↓0uε(x) is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.

AB - The Neumann problem with a small parameter (Formula presented.) is considered in this paper. The operators L0and L1are self-adjoint second order operators. We assume that L0has a non-negative characteristic form and L1is strictly elliptic. The reflection is with respect to inward co-normal unit vector γε(x). The behavior of limε↓0uε(x) is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.

KW - Averaging principle

KW - Diffusion processes on a graph

KW - Equations with non-negative characteristic form

KW - Second order equations with a small parameter

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DO - 10.1080/03605302.2013.812658

M3 - Article

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SN - 0360-5302

VL - 38

SP - 1712

EP - 1736

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

IS - 10

ER -

On Second Order Elliptic Equations with a Small Parameter

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