Previous |  Up |  Next

Article

Keywords:
divergence-free solutions; elliptic systems; systems of Stokes-type; regularity
Summary:
We give an example of a bounded discontinuous divergence-free solution of a linear elliptic system with measurable bounded coefficients in $\Bbb R^3$ and a corresponding example for a {\bf Stokes-like system}.
References:
[1] Campanato S.: Sistemi ellittici in forma divergenza. Regolarita' all'interno. Quaderno Sc. Norm. Sup. Pisa Pisa (1980). MR 0668196 | Zbl 0453.35026
[2] De Giorgi E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico. Boll. U.M.I. 4 (1968), 135-137. MR 0227827
[3] Frehse J.: Una generalizzazione di un controesempio di De Giorgi nella teoria delle equazioni ellitiche. Boll. U.M.I. 6 (1970), 998-1002. MR 0276612
[4] Galdi G.P.: An introduction to the mathematical theory of the Navier-Stokes equations, Volume I: Linearized steady problems. Springer tracts in natural philosophy vol.38, Springer-Verlag, New York Berlin Heidelberg et alii, 1994. MR 1284205
[5] Giaquinta M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Math. Studies, vol. 105 Princeton University Press Princeton (1983). MR 0717034 | Zbl 0516.49003
[6] Giaquinta M., Modica G.: Non linear systems of the type of the Navier-Stokes system. J. Reine Angewandte Math. 330 (1982), 173-214. MR 0641818
[7] John O., Malý J., Stará J.: Nowhere continuous solutions to elliptic systems. Comment. Math. Univ. Carolinae 30 (1989), 33-43. MR 0995699
[8] Koshelev A.I., Chelkak S.I.: Regularity of solutions of quasilinear elliptic systems. Teubner Leipzig (1985). MR 0825485 | Zbl 0581.35003
[9] Ladyzhenskaya O. A.: The mathematical theory of viscous incompressible flow. Gordon and Breach Science Publishers New York (1969). MR 0254401 | Zbl 0184.52603
[10] Leonardi S.: On constants of some regularity theorems. De Giorgi counterexamples. Preprint N.I.U (1993) [Thesis (1995-96)], to appear.
[11] Munkres J.R.: Analysis on manifolds. Addison-Wesley Redwood City (1991). MR 1079066 | Zbl 0743.26006
[12] Nečas J.: Introduction to the theory of nonlinear elliptic equations. Teubner, Leipzig, 1983 or J. Wiley, Chichester (1986). MR 0731261
[13] Nečas J.: Notes for a talk given at the Equadiff 8 in Bratislava 1993. (personal communication).
[14] Souček J.: Singular solutions to linear elliptic systems. Comment. Math. Univ. Carolinae 25 (1984), 273-281. MR 0768815
[15] Steinhauer M.: Funktionenräume vom Campanato-Typ und Regularitätseigenschaften elliptischer Systeme zweiter Ordnung. Diplomarbeit Univ. Bonn (1993).
Partner of
EuDML logo