Stability in the \(C^1\)-norm of classes of harmonic mappings (Q1288122)

The author calls a mapping \(f=(f_1,\ldots,f_m):U \to \mathbb R^n\) \(\varepsilon\)-quasiharmonic if it is continuously differentiable, belongs to the class \(W^2(U,\mathbb R^m)= \bigcup\limits_{p>n}W^2_{p,\text{loc}}(U,\mathbb R^m)\) and satisfies the differential inequality \[ | \Delta f(x)| \leq \varepsilon \biggl\{ n \sum_{i,j=1}^n| \partial_{i,j}f(x)^2| \biggr\}^{1/2} \] for almost all \(x\in U\). The notion of \(\xi^1\)-stability is proposed. Stability in the \(C^1\)-norm of classes of harmonic mappings with respect to \(\varepsilon\)-quasiharmonic mappings is studied on the basis of this notion, as \(\varepsilon \to 0\).