JournalszaaVol. 20, No. 2pp. 315–330

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

  • Jörg Wolf

    Humboldt-Universität zu Berlin, Germany
Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition cover
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Abstract

This paper is concerned with systems of nonlinear partial differential equations

Dαaiα(x,u,u)=bi(x,u,u)(i=1,...,N)–D_{\alpha}a^{\alpha}_i (x, u, \bigtriangledown u) = b_i (x, u, \bigtriangledown u) (i = 1,..., N)

where the coefficients aiαa^{\alpha}_i are assumed to satisfy the condition

aiα(x,u,ξ)aiα(y,v,ξ)ωxy+uv(1+ξ)a^{\alpha}_i (x, u, \xi) – a^{\alpha}_i (y, v, \xi) ≤ \omega |x–y| + |u–v| (1+|\xi|)

for all x,u,y,vΩ×RN{x, u}, {y, v} \in \Omega \times \mathbb R^N and all ξRnN\xi \in \mathbb R^{nN}, and where 01ω(t)tdt<+\int^1_0 \frac {\omega (t)}{t} dt < + \infty while the functions aiαξβj\frac {\partial a_i^{\alpha}}{\partial \xi ^j_{\beta}} satisfy the standard boundedness and ellipticity conditions and the function ξbi(x,u,ξ)\xi \mapsto b_i (x, u, \xi) may have quadratic growth. With these assumptions we prove partial Hölder continuity of bounded weak solutions uu to the above system provided the usual smallness condition on uL(Ω)\|u \|_{L \infty ({\Omega})} is fulfilled.

Cite this article

Jörg Wolf, Partial Regularity of Weak Solutions to Nonlinear Elliptic Systems Satisfying a Dini Condition. Z. Anal. Anwend. 20 (2001), no. 2, pp. 315–330

DOI 10.4171/ZAA/1018