A coerciveness inequality for a class of nonelliptic operators of constant deficit

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

$$\Lambda = - iE(x)^{ - 1} \sum\limits_{j = j}^n {A_j D_j } $$

. Here x=(x1, x2, ..., xn) ∈ Rn, Dj=α/αxj and E(x), A1, A2, ..., An are m×m Herminat matrices with the properties that A1, A2,...,An are constant, E(x) is uniformly positive definite on Rn and E(x) and the derivatives D1E(x), D2E(x),..., DnE(x) are continuous and bounded on Rn. 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

$$\sum\limits_{j = 1}^n {\left\| {D_j u} \right\|^2< K(\left\| {\Lambda u} \right\|^2 + \left\| u \right\|^2 )} for{\mathbf{ }}all{\mathbf{ }}u \in D(\Lambda ) \cap N(\Lambda )^ \bot $$

. In this paper it is shown that Λ is coercive on N(Λ) provided that lim E(x)=E0 uniformly in x/|x|, where E0 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∈Rn−{0} and x∈Rn. 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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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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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).

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