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Partial regularity of weak solutions to nonlinear elliptic systems satisfying a Dini condition. (English) Zbl 1163.35329

This paper deals with systems of nonlinear partial differential equations
\[ -D_\alpha a_i^\alpha(x,u,\nabla u)= b_i(x,u,\nabla u), \quad i=1,\dots,N, \]
where on the coefficients \(a_i^\alpha\) imposed following condition
\[ \big|a_i^\alpha(x,u,\xi)- a_i^\alpha(y,v,\xi)\big|\leq \omega\big(|x-y|+|u-v|\big) \big(1+|\xi|\big) \]
for all \((x,u),(y,v)\in\Omega\times\mathbb R^N\), with \(\Omega\) be a bounded domain and \(\int_0^1 \frac{\omega(t)}{t}\,dt<+\infty\).
Under certain additional conditions on \(a_i^\alpha\), \(b_i\) together with ellipticity conditions the author proves partial Hölder continuity of bounded weak solutions \(u\) to the above system provided that \(\|u\|_{L^\infty(\Omega)}\) is sufficiently “small”.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35K65 Degenerate parabolic equations
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