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Homogenization of periodic linear degenerate PDEs. (English) Zbl 1167.60015
Consider a second-order lindar partial differential operator \[ L_\epsilon := \frac 12 \sum_{i,j=1}^d a_{ij}(x/\epsilon)\partial_i\partial_j + \sum_{i=1}^d \left[\epsilon^{-1}b_i(x/\epsilon)+c_i(x/\epsilon)\right]\partial_i \] where the matrix \((a_{ij})_{ij}\) may degenerate (even vanish) in some open subset \(D\subset\mathbb{R}^d\). The authors study the behaviour as \(\epsilon\to 0\) of the following elliptic and parabolic PDEs: \[ L_\epsilon u^\epsilon(x) + f(x,x/\epsilon)u^\epsilon(x) = 0,\; x\in D, \quad u^\epsilon(x)=g(x),\;x\in\partial D \] resp. \[ \partial_t u^\epsilon(t,x) = L_\epsilon u^\epsilon(t,x) + \left(\epsilon^{-1} e(x(\epsilon)+ f(x,x/\epsilon)\right)u^\epsilon(t,x), \quad u^\epsilon(0,x)=g(x),\;x\in\mathbb{R}^d. \] It is assumed that \(f\) is bounded from above and periodic in the second argument and that \(g\) is continuous and polynomially bounded; moreover \(e\) is periodic. Using the fact that the solution of such systems can be expressed as functionals of certain stochastic processes arising as solution processes of stochastic (Itô) differential equations (the coefficients can be derived from the coefficients of the PDEs under investigation), the authors prove that \(u^\epsilon \to u\) for each \(x\) resp. \((t,x)\) as \(\epsilon\to 0\) where \(u\) is the solution of a suitably ‘homogenized’ PDE. The main innovation in these results is the high degree of degeneracy which is allowed in the authors’ approach. The paper also contains a study of the image of the homogenised diffusion matrix and, in the end, some worked examples.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
60J65 Brownian motion
60H07 Stochastic calculus of variations and the Malliavin calculus
35J25 Boundary value problems for second-order elliptic equations
35K15 Initial value problems for second-order parabolic equations
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