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Approximation theory methods for the study of diffusion equations. (English) Zbl 0842.41020

Müller, Manfred W. (ed.) et al., Approximation theory. Proceedings of the 1st international Dortmund meeting IDoMAT 95 held in Witten, Germany, March 13-17, 1995. Berlin: Akademie Verlag. Math. Res. 86, 9-26 (1995).
The paper deals with a degenerate elliptic second order differential operator \(A: D(A)\to {\mathcal C} (\overline {\Omega})\) of the form \[ Au (x)= \sum^p_{i,j=1} \alpha_{ij} (x) {{\partial^2 u(x)} \over {\partial x_i \partial x_j}}+ \sum^p_{i=1} \beta_i (x) {{\partial u(x)} \over {\partial x_i}} \] for every \(u\in D(A)\) and \(x\in \Omega\). Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^p\), \(p\geq 1\) and \({\mathcal C} (\overline {\Omega})\) is the Banach lattice of all real-valued continuous functions defined on \(\overline {\Omega}\). The author introduces the closable operator \((A, D(A))\) so that its closure \((\overline {A}, D(\overline {A}))\) is the generator of a strongly continuous semigroup \((T(t) )_{t\geq 0}\) in \({\mathcal C} (\overline {\Omega})\). Then the Cauchy problem \[ {{\partial u(x, t)} \over {\partial t}}= \overline {A} (u(\cdot, t)) (x) \quad (x\in \Omega,\;t\geq 0), \qquad u(x,0)= u_0 (x) \quad (x\in \overline {\Omega},\;u_0\in D(\overline {A} )), \] has a unique solution given by \(u(x, t)= (T(t) u_0) (x)\) \((x\in \overline {\Omega}\), \(t\geq 0)\), and the solution continuously depends on the initial data \(u_0\). The author’s aim is to investigate the possibility of constructing a suitable approximation process \((T_n )_{n\in \mathbb{N}}\) of linear operators on \({\mathcal C} (\overline {\Omega})\) (i.e. \(\lim_{n\to\infty} T_n (f)= f\)) uniformly on \(\overline {\Omega}\) for every \(f\in C(\overline {\Omega}))\) such that for every \(t\geq 0\) and \(f\in {\mathcal C} (\overline {\Omega})\) \(T(t) f=\lim_{n\to \infty} T_n^{k (n)} f\) uniformly on \(\overline {\Omega}\) for a suitable sequence \((k(n) )_{n\geq 1}\) of positive integers, where each \(T_n^{(k)}\) denotes the power of \(T_n\) of order \(k(n)\). The basic methods of the approach together with a survey of the main results which have been obtained, are represented in the article.
For the entire collection see [Zbl 0830.00034].
Reviewer: I.E.Tralle (Minsk)

MSC:

41A35 Approximation by operators (in particular, by integral operators)
35J25 Boundary value problems for second-order elliptic equations
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