Mathematical Notes

, Volume 64, Issue 2, pp 220–229

# Mean value theorems for solutions of linear partial differential equations

• A. V. Pokrovskii
Article

## Abstract

We consider generalized mean value theorems for solutions of linear differential equations with constant coefficients and zero right-hand side which satisfy the following homogeneity condition with respect to a given vectorM with positive integer components: for each partial derivative occurring in the equation, the inner product of the vector composed of the orders of this derivative in each variable by the vectorM is independent of the derivative. The main results of this paper generalize the well-known Zalcman theorem. Some corollaries are given.

### Key words

partial differential equations measure space of distributions mean value theorem weighted homogeneity

## Preview

### References

1. 1.
L. Zalcman, “Mean values and differential equations,”Israel J. Math.,14, 339–352 (1973).
2. 2.
W. Fulks, “A mean value theorem for the heat equation,”Proc. Amer. Math. Soc.,17, 6–11 (1966).
3. 3.
L. P. Kuptsov, “About the mean value property for the heat equation,”Mat. Zametki [Math. Notes],29, No. 2, 211–222 (1981).
4. 4.
L. Hörmander,The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, Heidelberg (1983).Google Scholar
5. 5.
L. Hörmander,The Analysis of Linear Partial Differential Operators, Vol. 2, Springer, Heidelberg (1983).Google Scholar
6. 6.
L. Hörmander,An Introduction to Complex Analysis in Several Variables, Toronto (1966).Google Scholar
7. 7.
B. van der Waerden,Algebra [Russian translation], Nauka, Moscow (1979).Google Scholar
8. 8.
L. R. Volevich, “Local properties of solutions to quasielliptic systems,”Mat. Sb. [Math. USSR-Sb.],59 (101), 3–52 (1962).
9. 9.
R. Edwards,Functional Analysis. Theory and Applications New York (1965).Google Scholar
10. 10.
S. M. Nikol'skii,Approximation Theory for Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1977).Google Scholar
11. 11.
V. V. Grushin, “Relation between local and global properties of solutions to hypoelliptic partial differential equations,”Mat. Sb. [Math. USSR-Sb.],66 (108), No. 4, 525–550 (1965).