Riv. Mat. Univ. Parma, Vol. 5, No. 2, 2014

Taku Kanazawa[1]

Partial regularity for elliptic systems with VMO-coefficients

Pages: 311-333
Received: 15 February 2013
Accepted in revised form: 30 July 2013
Mathematics Subject Classification (2010): 35J60, 35B65.

Keywords: Nonlinear elliptic systems, Partial regularity, VMO-coefficients, $$\mathcal{A}$$-harmonic approximation.
Author address:
[1] : Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

Abstract: We establish partial Hölder continuity for vector-valued solutions $$u:\Omega\to\mathbb{R}^N$$ to inhomogeneous elliptic systems of the type: $-\mathrm{div}(A(x,u,Du))=f(x,u,Du) \qquad \mathrm{in} \>\Omega ,$ where the coefficients $$A:\Omega\times\mathbb{R}^N\times{\mathrm{Hom}}(\mathbb{R}^n,\mathbb{R}^N)\to{\mathrm{Hom}}(\mathbb{R}^n,\mathbb{R}^N)$$ are possibly discontinuous with respect to $$x$$. More precisely, we assume a VMO-condition with respect to the $$x$$ and continuity with respect to $$u$$ and prove Hölder continuity of the solutions outside of singular sets.

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