Differential and Integral Equations

A Trotter-type theorem for nonlinear stochastic equations in variational formulation and homogenization

Ioana Ciotir

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Abstract

This paper is concerned with the nonlinear partial differential equations of calculus of variations perturbed by noise in the Gelfand triple $V\subset H\subset V^{\prime }$. The main result is a Trotter-type theorem for this equation. In the second part of the paper we prove that, if we assume graph convergence of the sequence of nonlinear operators $ \{ A^{\alpha} \} _{\alpha }$, we have convergence of the corresponding sequence of invariant measures. Those results are used in the last part of the paper to study the homogenization problem for the equation, in the case of the differential operator of the type $A ( u ) =- \text{\mathrm {div}} [ a ( \nabla u ) ], $ for $u\in H_{0}^{1} ( \mathcal{O} ).$

Article information

Source
Differential Integral Equations, Volume 24, Number 3/4 (2011), 371-388.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019037

Mathematical Reviews number (MathSciNet)
MR2757465

Zentralblatt MATH identifier
1240.60178

Subjects
Primary: 35K57: Reaction-diffusion equations 60J60: Diffusion processes [See also 58J65] 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]

Citation

Ciotir, Ioana. A Trotter-type theorem for nonlinear stochastic equations in variational formulation and homogenization. Differential Integral Equations 24 (2011), no. 3/4, 371--388. https://projecteuclid.org/euclid.die/1356019037


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