Annali di Matematica Pura ed Applicata

, Volume 104, Issue 1, pp 31–42

# A function theoretic method forΔ 4 2 u+Q(x)u=0

• R. P. Gilbert
• D. Kukral
Article

## Summary

The approach used in this paper generalizes Colton's treatment [4] of certain second order elliptic equations in four independent variables to the fourth order case. This method is essentially a function theoretic one that is based on the earlier work of Tjong [11]. An integral operator is found that permits one to construct a complete family of solutions with respect to uniform convergence in compact sets of R4. Consequently, one is provided with a useful numerical procedure for solving the associated boundary value problems.

## Keywords

Integral Operator Elliptic Equation Fourth Order Uniform Convergence Numerical Procedure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
S. Bergman,Integral Operators in the Theory of Linear Partial Differential Equations, Engebnisse der Mathematik und ihrer Grenzgebiete, N. F., Heft 23, Springer-Verlag, Berlin, 1961, MR 25, no. 5277.
2. [2]
D. Colton,Integral operators for elliptic equations in three independent variables, I, II, Applicable Analysis (to appear).Google Scholar
3. [3]
D. Colton,Bergman Operators for elliptic equations in three independent variables, Bulletin Amer. Math. Soc.,77 (1971), pp. 752–756.
4. [4]
D. Colton,Bergman operators for elliptic equations in four independent variables, SIAM J. Math. Anal.,3 (1972), pp. 401–412.
5. [5]
D. Colton -R. P. Gilbert,An integral operator approach to Cauchy's problem for Δ p+2 u=0, SIAM J. Math. Anal.,2 (1971), pp. 113–132.
6. [6]
R. P. Gilbert,Function Theoretic Methods in Partial Differential Equations, Academic Press, New York, 1969.
7. [7]
R. P. Gilbert - D. Kukral,A function theoretic method for Δ 32 u+(x)u=0 (to appear).Google Scholar
8. [8]
R. P. Gilbert -C. Y. Lo,On the approximation of solutions of elliptic partial differentia equations in two and three dimensions, SIAM J. Math. Anal.,2 (1971), pp. 17–30.
9. [9]
L. Hörmander,Linear Partial Differential Operators, Springer-Verlag, Berlin, 1964.Google Scholar
10. [10]
D. Kukral,Constructive Methods for Determining the Solutions of Higher Order Elliptic Partial Differential Equations, Ph. D. Thesis, Indiana University, 1972.Google Scholar
11. [11]
B. L. Tjong,Operators generating solutions of Δ 3 ψ+Fψ=0 and their properties, Analytic Methods in Math. Physics, Gordon and Breach, New York, 1970.Google Scholar
12. [12]
I. N. Vekua,New Methods for solving Elliptic Equations, OGI, Moscow, 1948; Eng. transl., Series in Appl. Math., vol.1, North-Holland, Amsterdam; Interscience, New York, 1967; MR11, 598; MR35, no. 3243.Google Scholar