Fritz John pp 425-459

# Continuous Dependence on Data for Solutions of Partial Differential Equations With a Prescribed Bound

• Fritz John
Chapter
Part of the Contemporary Mathematicians book series (CM)

## Abstract

Problems in partial differential equations usually require that a solution u be determined from certain data f. Ordinarily the data consist of the values of u and of a certain number of derivatives of u on a manifold Φ. For the classical problems which are well posed in the sense of Hadamard the solution u exists and is determined uniquely for all f of some class C s ; moreover u depends continuously on f if suitable norms are used. Usually no solution u exists in a problem which is not well posed, even for f in C. Moreover, even for those f for which u exists we have no continuous dependence of u on f. This is best illustrated by the classical example of Hadamard [1] of the Cauchy problem for the Laplace equation u xx +u=0, u=0, u y =f(x), for which no solution exists unless f is analytic. If f(x) = n -2 cos nx we do have a solution u = n -3 (cos nx)(sinh ny). However though f and its first and second derivatives tend to zero for n →∞ the corresponding solutions u tend to ∞ for x = 0, y ≠ 0.

## Bibliography

1. [1]
Åsgeirsson, L., Über Mittelwertgleichungen, die mehreren partiellen Differentialgleichungen 2er Ordnung zugeordnet sind, Studies and Essays, Courant Anniversary Vol., Interscience Publishers, New York, 1948.Google Scholar
2. [2]
Courant, R., and Hilbert, D., Methoden der mathematischen Physik, Vol. 2, Springer, Berlin, 1937.
3. [3]
Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, 1923.Google Scholar
4. [4]
Holmgren, E., Über Systeme von linearen partiellen Differentialgleichungen, Ofversigt af kongl. Vetenskaps-Akademiens Förhandlingar, Vol. 58, 1901, pp. 91–103.Google Scholar
5. [5]
Hörmander, L., On the theory of general partial differential operators, Acta Math., Vol. 94, 1955, pp. 161–248.
6. [6]
John, F., Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience Publishers, New-York, 1955.Google Scholar
7. [7]
John, F., On linear partial differential equations with analytic coefficients. Unique continuation of data, Comm. Pure Appl. Math., Vol. 2, 1949, pp. 209–253.
8. [8]
John, F., Numerical solution of problems which are not well posed in the sense of Hadamard, Proc. Rome Symposium, Provisional International Computation Centre, January, 1959, pp. 103–116.Google Scholar
9. [9]
Landis, E. M., On some properties of solutions of elliptic equations, Dolkady Akad. Nauk SSSR N.S., No. 5, Vol. 107, 1956, pp. 640–643.Google Scholar
10. [10]
Nirenberg, L., Uniqueness of Cauchy problems for differential equations with constant leading coefficients, Comm. Pure Appl. Math., Vol. 10, 1957, pp. 89–105.Google Scholar
11. [11]
Nirenberg, L., Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math., Vol. 9, 1956, pp. 509–529.Google Scholar
12. [12]
Schwartz, L., Théorie des Distributions, 2nd Ed., Hermann et Cie., Paris, 1957.Google Scholar
13. [13]
Sobolev, S. L., On a theorem of functional analysis, Mat. Sbornik, N.S. Vol. 4, 1938, pp. 471–497.Google Scholar
14. [14]
Watson, G. N., A Treatise on the Theory of Bessel Functions, 2nd Ed., The University Press, Cambridge, 1945.Google Scholar