Annali di Matematica Pura ed Applicata (1923 -)

, Volume 116, Issue 1, pp 17–56 | Cite as

Quasilinear elliptic systems

  • A. V. Lair


The quasilinear elliptic system
$$\sum\limits_{l{\text{ = 1}}}^n {\frac{\partial }{{\partial x_l }}\left\{ {\sum\limits_{j = 1}^N {\sum\limits_{m = 1}^n {C_{ij}^{lm} [x,U]\frac{{\partial U^j }}{{\partial x_m }} + B_i^l [x,U]} } } \right\} + F_i [x,U] = 0} $$

1⩽i⩽N, x in a bounded domain Ω, and U=0 on the boundary of Ω is studied. Under various assumptions regarding the auxiliary functions C, B, and F, the author studies weak existence, uniqueness, and stability in H 0 1 (Ω). In addition, by requiring C ij lm =0 for i ≠ j, it is proved that such weak solutions have bounded L(Ω) norm and satisfy a Hölder condition on the closure of Ω.


Weak Solution Bounded Domain Auxiliary Function Elliptic System Author Study 


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  1. [1]
    L. BersF. JohnM. Schechter,Partial Differential Equations, Wiley (Interscience), New York, 1964.zbMATHGoogle Scholar
  2. [2]
    F. E. Browder,Functional analysis and partial differential equations — I, Ann. of Math.,138 (1959), pp. 55–79.MathSciNetCrossRefGoogle Scholar
  3. [3]
    F. E. Browder,Existence theory of boundary value problems for quasilinear elliptic systems with strongly nonlinear lower order terms, Partial Differential Equations (Proc. Sympos. Pure Math., vol. XXIII, Univ. of Calif., Berkeley, Calif., 1971), Amer. Math. Soc., Providence (1973), pp. 269–286.Google Scholar
  4. [4]
    J. R. CannonW. T. FordA. V. Lair,Quasilinear parabolic systems, J. Diff. Eqs.20 (1976), pp. 441–472.MathSciNetCrossRefGoogle Scholar
  5. [5]
    W. T. Ford,On the first boundary value problem for [h(x, x′, t)]′ = f(x, x′, t), Proc. of the Amer. Math. Soc.,35 (1972), pp. 491–497.MathSciNetzbMATHGoogle Scholar
  6. [6]
    G. H. HardyJ. E. LittlewoodG. Polya,Inequalities, 2nd ed., Cambridge Univ. Press, New York, 1952.zbMATHGoogle Scholar
  7. [7]
    A. V. Ivanov,The solvability of the Dirichlet problem for certain classes of second order elliptic systems, Pric. of the Steklov Inst. Math.,125 (1973), pp. 49–79.MathSciNetGoogle Scholar
  8. [8]
    O. A. LadyzhenskayaN. N. Ural'tseva,Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.zbMATHGoogle Scholar
  9. [9]
    A. V. Lair,A Rellich compactness theorem for sets of finite volume, Amer. Math. Monthly83 (1976), pp. 350–351.MathSciNetCrossRefGoogle Scholar
  10. [10]
    C. Miranda,Partial Differential Equations of Elliptic Type, 2nd rev. ed., Springer-Verlag, New York, 1970.zbMATHGoogle Scholar
  11. [11]
    H. L. Royden,Real Analysis, 2nd ed., MacMillan, New York, 1968.zbMATHGoogle Scholar
  12. [12]
    J. T. Schwartz,Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.zbMATHGoogle Scholar
  13. [13]
    M. I. Visik,Quasi-linear strongly elliptic systems of differential equations in divergence form, Trans. Moscow Math. Soc., 1963, pp. 148–208.Google Scholar
  14. [14]
    K. Yosida,Functional Analysis, 2nd ed., Springer-Verlag, New York, 1968.CrossRefGoogle Scholar

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© Fondazione Annali di Matematica Pura ed Applicata (1923 -) 1978

Authors and Affiliations

  • A. V. Lair
    • 1
  1. 1.University of South DakotaVermillion

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