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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 2, 2001, pp. 315–330
DOI: 10.4171/ZAA/1018

Published online: 2001-06-30

Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition

Jörg Wolf[1]

(1) Humboldt-Universität zu Berlin, Germany

This paper is concerned with systems of nonlinear partial differential equations $$–D_{\alpha}a^{\alpha}_i (x, u, \bigtriangledown u) = b_i (x, u, \bigtriangledown u) (i = 1,..., N)$$ where the coefficients $a^{\alpha}_i$ are assumed to satisfy the condition $$a^{\alpha}_i (x, u, \xi) – a^{\alpha}_i (y, v, \xi) ≤ \omega |x–y| + |u–v| (1+|\xi|)$$ for all ${x, u}, {y, v} \in \Omega \times \mathbb R^N$ and all $\xi \in \mathbb R^{nN}$, and where $\int^1_0 \frac {\omega (t)}{t} dt < + \infty$ while the functions $\frac {\partial a_i^{\alpha}}{\partial \xi ^j_{\beta}}$ satisfy the standard boundedness and ellipticity conditions and the function $\xi \mapsto b_i (x, u, \xi)$ may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions $u$ to the above system provided the usual smallness condition on $\|u \|_{L \infty ({\Omega})}$ is fulfilled.

Keywords: Nonlinear elliptic systems, partial regularity, blow-up method

Wolf Jörg: Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition. Z. Anal. Anwend. 20 (2001), 315-330. doi: 10.4171/ZAA/1018