# Quasilinear elliptic systems

Article

First Online:

- 55 Downloads

## Summary

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.

## Keywords

Weak Solution Bounded Domain Auxiliary Function Elliptic System Author Study## Preview

Unable to display preview. Download preview PDF.

## References

- [1]L. Bers —F. John —M. Schechter,
*Partial Differential Equations*, Wiley (Interscience), New York, 1964.zbMATHGoogle Scholar - [2]F. E. Browder,
*Functional analysis and partial differential equations — I*, Ann. of Math.,**138**(1959), pp. 55–79.MathSciNetCrossRefGoogle Scholar - [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]J. R. Cannon —W. T. Ford —A. V. Lair,
*Quasilinear parabolic systems*, J. Diff. Eqs.**20**(1976), pp. 441–472.MathSciNetCrossRefGoogle Scholar - [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]G. H. Hardy —J. E. Littlewood —G. Polya,
*Inequalities*, 2nd ed., Cambridge Univ. Press, New York, 1952.zbMATHGoogle Scholar - [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]O. A. Ladyzhenskaya —N. N. Ural'tseva,
*Linear and Quasilinear Elliptic Equations*, Academic Press, New York, 1968.zbMATHGoogle Scholar - [9]A. V. Lair,
*A Rellich compactness theorem for sets of finite volume*, Amer. Math. Monthly**83**(1976), pp. 350–351.MathSciNetCrossRefGoogle Scholar - [10]C. Miranda,
*Partial Differential Equations of Elliptic Type*, 2nd rev. ed., Springer-Verlag, New York, 1970.zbMATHGoogle Scholar - [11]
- [12]J. T. Schwartz,
*Nonlinear Functional Analysis*, Gordon and Breach, New York, 1969.zbMATHGoogle Scholar - [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]

## Copyright information

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