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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