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Zbl 1150.35385
Kronz, Manfred
Quasimonotone system of higher order. (English)
[J] Boll. Unione Mat. Ital., Sez. B, Artic. Ric. Mat. (8) 6, No. 2, 459-480 (2003). ISSN 0392-4041

The technique of (poly)harmonic approximation is applied to derive a partial regularity result for quasimonotone nonlinear systems of higher order in divergence form of the type $$\int A(x,D^mu)D^m\varphi \,dx=0$$ for all test functions $\varphi$. Quasimonotonicity is a weak sort of ellipticity condition. \par Assuming natural growth conditions on $A$ and Dini continuity of $A$ with respect to $x$, it is proved that weak solutions of the system have continuous $m$-th derivatives (with modulus of continuity derived from the Dini condition) outside a closed singular set of vanishing Lebesgue measure. Moreover, the singular set can be characterized by some Lebesgue-point type condition and the vanishing of some $m$th order excess. \par The corresponding result for second order systems, which is generalized to arbitrary order here, has been proven by {\it C. Hamburger} [Ann. Mat. Pura Appl. (4) 169, 321--354 (1995; Zbl 0852.35031)].
[Andreas Gastel (Erlangen)]

MSC 2000:: *35J45 Systems of elliptic equations, general
35G20 General theory of nonlinear higher-order PDE
35D10 Regularity of generalized solutions of PDE

Citations: Zbl 0852.35031

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