# Quasimonotonicity, regularity and duality for nonlinear systems of partial differential equations

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## Summary

We prove partial regularity for the vector-valued differential forms solving the system δ(A(x, ω))=0, dω=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) ≈ (1+ + ¦ω¦^{2})^{(p − 2)/2} ω (p⩾2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, ω)=D_{ω}f(x, ω) of a real-valued function*f*, is analogous to but stronger than quasiconvexity for f. The case 1<p<2 for monotone A is reduced to the case p⩾2 by a duality technique.

## Keywords

Differential Equation Partial Differential Equation Nonlinear System Differential Form Partial Regularity
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## References

- [A-F]Acerbi E. -Fusco N.,
*Regularity for minimizers of non-quadratic functionals: the case*1<p<2, J. Math. Anal. Appl.,**140**(1989), pp. 115–135.Google Scholar - [B]Ball J. M.,
*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rational Mech. Anal.,**63**(1977), pp. 337–403.Google Scholar - [C1]Campanato S.,
*Differentiability of the solutions of nonlinear elliptic systems with natural growth*, Ann. Mat. Pura AppL,**131**(1982), pp. 75–106.Google Scholar - [C2]Campanato S.,
*Hölder continuity of the solutions of some non-linear elliptic systems*, Adv. Math.,**48**(1983), pp. 16–43.Google Scholar - [Ci]Ciarlet P. G.,
*Mathematical elasticity, Volume I: Three-dimensional elasticity*, North-Holland, Amsterdam, New York, Oxford (1988).Google Scholar - [DS]Duff G. F. D. -Spencer D. C.,
*Harmonic tensors on Riemannian manifolds with boundary*, Ann. Math.,**56**(1952), pp. 128–156.Google Scholar - [E]Evans L. C.,
*Quasiconvexity and partial regularity in the calculus of variations*, Arch. Rational Mech. Anal,**95**(1986), pp. 227–252.Google Scholar - [F]Fuchs M.,
*Regularity theorems for nonlinear systems of partial differential equations under natural ellipticity conditions*, Analysis,**7**(1987), pp. 83–93.Google Scholar - [G]Giaquinta M.,
*Multiple integrals in the calculus of variations and nonlinear elliptic systems*, Princeton Univ. Press, Princeton (1983).Google Scholar - [G-G]Giaquinta M. -Giusti E.,
*On the regularity of the minima of variational integrals*, Acta Math.,**148**(1982), pp. 31–46.Google Scholar - [G-M1]Giaquinta M. -Modica G.,
*Almost-everywhere regularity results for solutions of nonlinear elliptic systems*, Manuscripta Math.,**28**(1979), pp. 109–158.Google Scholar - [G-M2]Giaquinta M. -Modica G.,
*Partial regularity of minimizers of quasiconvex integrals*, Ann. Inst. H. Poincaré Anal. Non Linéaire,**3**(1986), pp. 185–208.Google Scholar - [G-S]Giaquinta M. -Souček J.,
*Caccioppoli's inequality and Legendre-Hadamard condition*, Math. Ann.,**270**(1985), pp. 105–107.Google Scholar - [G-T]Gilbarg G. -Trudinger N. S.,
*Elliptic partial differential equations of second order*, Springer, Berlin, Heidelberg, New York (1983).Google Scholar - [H]Hamburger C.,
*Regularity of differential forms minimizing degenerate elliptic functionals*, J. Keine Angew. Math.,**431**(1992), pp. 7–64.Google Scholar - [I]Ivert P.-A.,
*Regularitätsuntersuchungen von Lösungen elliptischer Systeme von quasilinearen Differentialgleichungen zweiter Ordnung*, Manuscripta Math.,**30**(1979), pp. 53–88.Google Scholar - [M]Morrey C. B. Jr.,
*Multiple integrals in the calculus of variations*, Springer, Berlin, Heidelberg, New York (1966).Google Scholar - [N-W]Naumann J. -Wolf J.,
*On the interior regularity of weak solutions of degenerate elliptic systems (the case 1<p<2)*, Rend. Sem. Mat. Univ. Padova,**88**(1992), pp. 55–81.Google Scholar - [Z]Zhang Ke-Wei,
*On the Dirichlet problem for a class of quasilinear elliptic systems of partial differential equations in divergence form*, In:Chern S. S. (ed.),*Partial differential equations, Proceedings*,*Tianjin 1986*, (Lecture Notes in Mathematics,**1306**, pp. 262–277) Springer, Berlin, Heidelberg, New York (1988).Google Scholar

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