Stability in the \(C^l\)-norm of classes of solutions to systems of linear partial differential equations of elliptic type
Consider an elliptic system \[ D_jg(x)=\sum\limits_{\varkappa=1}^m\sum\limits_{|p|=l} a_p^{j\varkappa}\partial^pg_\varkappa (x)=0, \qquad j=1,\dots ,k, \] of linear partial differential equations of order \(l\) with constant coefficients. Put \(D=(D_1,\dots ,D_k)\). Let \(\mathfrak G_D\) be the class of all solutions to the system \(Dg=0\). Roughly speaking, \(\mathfrak G_D\) is said to be stable in the \(C^l\)-norm if there exists a function \(\alpha: [0,+\infty)\to[0,+\infty)\) such that \(\lim\limits_{\varepsilon\to 0}\alpha (\varepsilon)=\alpha (0)=0\) and, for every (generalized) solution \(f\) to the differential inequality \(|Df(x)|\leq \varepsilon \|f^{(l)}(x)\|\), there exists a mapping \(g\in\mathfrak G_D\) such that, for every \(u=0,\dots ,l-1\), the \(C^u\)-norm of the mapping \(f-g\) is less than or equal to \(\alpha (\varepsilon)\). Historically, this notion is inspired by the notion of stability in the \(C\)-norm which came from the theory of quasiconformal mappings [see, for example, the book by \textit{Yu.~G.~Reshetnyak}, ``Stability theorems in geometry and analysis'' (1996; Zbl 0848.30013)]. The author created a general stability theory of classes of mappings in the \(C\)-norm and the \(C^1\)-norm in his previously published numerous papers [see, for example, his book ``Stability in the \(C\)-norm of classes of mappings'' (1990; Zbl 0772.30023) and his recent article in Sib. Mat. Zh. 39, No. 6, 1304-1321 (1998; Zbl 0916.35006)]. In the article under review, stability of classes \(\mathfrak G_D\) is studied in the \(C^l\)-norm for arbitrary \(l=0,1,2,\dots \) .
scientific article; zbMATH DE number 1432155