## Summary

This paper presents a coerciveness inequality for a class of nonelliptic first-order matrix partial differential operators of the form

. Here x=(x_{1}, x_{2}, ..., x_{n}) ∈ R^{n}, D_{j}=α/αx_{j} and E(x), A_{1}, A_{2}, ..., A_{n} are m×m Herminat matrices with the properties that A_{1}, A_{2},...,A_{n} are constant, E(x) is uniformly positive definite on R^{n} and E(x) and the derivatives D_{1}E(x), D_{2}E(x),..., D_{n}E(x) are continuous and bounded on R^{n}. Λ operates on functions u(x) whose values are m × 1 matrices. E(x) determines a Hilbert space ℋ of such functions with inner product\((u, v)_E = \int\limits_{R^n } {u(x)^ * E(x)v(x)dx} \) and Λ is essenially selfadjoint on ℋ. Λ is said to be coercive on N(Λ)^{⊥}, the orthogonal complement in ℋ of the null space N(Λ), if there exists a constant K>0 such that

. In this paper it is shown that Λ is coercive on N(Λ)^{⊤} provided that lim E(x)=E_{0} uniformly in x/|x|, where E_{0} is positive definite, and the symbol\(\Lambda (p, x) = E(x)^{ - 1} \sum\limits_{j = 1}^n {A_j p_j } \) satisfies rank Λ(p, x)=m−k for all p∈R^{n}−{0} and x∈R^{n}. Such operators are said to have constant deficit k. Λ is elliptic if and only if it has constant deficit k=0.

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## Additional information

Entrata in Redazione il 19 aprile 1971.

This research was supported by the U.S. Office of Naval Research and by the Swiss National Fund for Scientific Research. Reproduction in whole or part is permitted for any purpose of the U.S. Government.

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### Cite this article

Schulenberger, J.R., Wilcox, C.H. A coerciveness inequality for a class of nonelliptic operators of constant deficit.
*Annali di Matematica* **92, **77–84 (1972). https://doi.org/10.1007/BF02417937

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### Keywords

- Hilbert Space
- Differential Operator
- Null Space
- Orthogonal Complement
- Partial Differential Operator