**Taku Kanazawa ^{[1]}**

[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.

**References**

[1] V. Bogelein, F. Duzaar, J. Habermann and C. Scheven, *Partial Holder continuity for discontinuous elliptic problems with VMO-coefficients*, Proc. Lond. Math. Soc. (3) 103 (2011), 371-404.
[MR2827000]

[2] S. Campanato, *Equazioni ellittiche del \({\rm II}^\circ\) ordine e spazi \({\mathfrak L}^{(2,\lambda )}\)*, Ann. Mat.Pura Appl. (4) 69 (1965), 321-381.
[MR0192168]

[3] S. Campanato, *Holder continuity and partial Holder continuity results for \(H^{1,q}\)-solutions of nonlinear elliptic systems with controlled growth*, Rend. Sem. Mat. Fis. Milano 52 (1982), 435-472.
[MR0802957]

[4] S. Campanato, *Holder continuity of the solutions of some nonlinear elliptic systems*, Adv. in Math. 48 (1983), 16-43.
[MR0697613]

[5] S. Chen and Z. Tan, *Optimal interior partial regularity for nonlinear elliptic systems under the natural growth condition: the method of A-harmonic approximation*, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), 491-508.
[MR2339389]

[6] F. Duzaar and A. Gastel, *Nonlinear elliptic systems with Dini continuous coefficients*, Arch. Math. (Basel) 78 (2002), 58-73.
[MR1887317]

[7] F. Duzaar and J. F. Grotowski, *Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation*, Manuscripta Math. 103 (2000), 267-298.
[MR1802484]

[8] M. Foss and G. Mingione, *Partial continuity for elliptic problems*, Ann. Inst. H. Poincare Anal. Non Linéaire 25 (2008), 471-503.
[MR2422076]

[9] M. Giaquinta and L. Martinazzi, *An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs*, Appunti.
Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], vol. 2, Edizioni della Normale, Pisa 2005.
[MR2192611]

[10] M. Giaquinta and G. Modica, *Almost-everywhere regularity results for solutions of nonlinear elliptic systems*, Manuscripta Math. 28 (1979), 109-158.
[MR0535699]

[11] M. Giaquinta and G. Modica, *Partial regularity of minimizers of quasiconvex integrals*, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 185-208.
[MR0847306]

[12] M. Kronz, *Partial regularity results for minimizers of quasiconvex functionals of higher order*, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 81-112.
[MR1902546]

[13] Y. Qiu, *Optimal partial regularity of second order nonlinear elliptic systems with Dini continuous coefficients for the superquadratic case*, Nonlinear Anal. 75 (2012), 3574-3590.
[MR2901339]

[14] M. A. Ragusa and A. Tachikawa, *Regularity of minimizers of some variational integrals with discontinuity*, Z. Anal. Anwend. 27 (2008), 469-482.
[MR2448746]

[15] S. Z. Zheng, *Partial regularity for quasi-linear elliptic systems with VMO coefficients under the natural growth*, Chinese Ann. Math. Ser. A 29 (2008), 49-58.
[MR2397284]

Home Riv.Mat.Univ.Parma