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Boundary regularity for certain quasilinear elliptic systems of divergence structure. (English) Zbl 0777.35016
Summary: Consider the Dirichlet problem for a quasilinear elliptic system of the form $-D_ j\bigl(a^{ij}(x,u,Du)D_ i u^ k\bigr)=-D_ i f^ i_ k+g^ k,$ and suppose that the structural condition $-D_ j\bigl(a^{ij}(x,u,Du)\bigr)=h^ i\in L^ \infty$ is satisfied. Then $$\varepsilon$$-regularity is proven for $$Du$$, i.e., under a certain smallness condition any Lipschitz solution has a Hölder continuous gradient in $$\overline\Omega$$.
##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations
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##### References:
 [1] [C] Campanato, S.: Equazioni ellittiche del IIo ordine e spazi (2,$$\lambda$$). Ann. Mat. Pura Appl.69, 321–381 (1965) · Zbl 0145.36603 · doi:10.1007/BF02414377 [2] [GE] Gehring, F. W.: The LP-integrablity of the partial derivatives of a quasiconformal mapping. Acta Math.130, 265–277 (1973) · Zbl 0258.30021 · doi:10.1007/BF02392268 [3] [G] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann. of Math Studies105, Princeton, N. J.: Princeton University Press 1983 · Zbl 0516.49003 [4] [GG] Giaquinta, M. and E. Giusti: Nonlinear elliptic systems with quadratic growth. Manuscripta Math.24, 323–349 (1978) · Zbl 0378.35027 · doi:10.1007/BF01167835 [5] [GM] Giaquinta, M. and G. Modica: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math.311/312, 149–169 (1979) · Zbl 0409.35015 [6] [M] Morrey C. B.: Second order elliptic systems of differential equations. In: Ann. of Math. Studies33, Princeton, N. J.: Princeton University Press 1954, pp. 101–159 · Zbl 0057.08301 [7] [S1] Schulz, F.: Über nichtlineare, konkave elliptische Differentialgleichungen. Math. Z.191, 429–448 (1986 · Zbl 0594.35033 · doi:10.1007/BF01162718 [8] [S2] Schulz, F.: Regularity for certain quasilinear elliptic systems of divergence structure. Indiana Univ. Math. J.39, 303–314 (1990) · Zbl 0712.35027 · doi:10.1512/iumj.1990.39.39017 [9] [S3] Schulz, F.: Boundary regularity for certain fully nonlinear elliptic equations. J. Reine Angew. Math. (to appear)
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