Abstract
This paper deals with first-order matrix partial differential operators of the form
where x=(x1, ..., xn)∈Rn, Dj=∂/∂xj, and the Lj(x), j=0, 1, 2, ..., n, are m′×m matrix-valued functions of x. Let L 2, m , be the Hilbert space of square integrable m-vector-valued functions on Rn. The operator(1) determines a closed linear operator L : L 2, m →L 2, m′ . L is said to be coercive on a subspace V⊂L 2, m if there is a constant μ>0 such that
for all u∈D(L)∩V, where D(L) denotes the domain of L and ∥·∥ denotes the norm in L2, m. L is said to have constant deficit k in Rn if the symbol\(L(p,x) = \mathop \sum \limits_{j = 1}^n L_j (x)p_j\) has constant rank m–k for all x∈Rn and p∈Rn−{0} (L is elliptic if and only if k=0). The paper gives criteria for nonelliptic operators L of constant deficit k to be coercive on subspaces. In particular, operators of the form\(\Lambda = - iE(x)^{ - 1} \mathop \sum \limits_{j = 1}^n A_j D_j\) are considered where E(x) and Aj are m×m Hermitian matrices and E(x) is positive definite.Λ defines a self-adjoint operator on the Hilbert space H with inner product\((u,v)E = \mathop \smallint \limits_{R^n } u^* Ev dx\). It is shown, under suitable hypotheses on E(x) and Aj, thatΛ is coercive on the subspace N(Λ)⊥, the orthogonal complement in H of N(Λ), the nullspace ofΛ.
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This research was supported by the Office of Naval Research. Reproduction in whole or in part is permitted for any purpose of the United States Government.
Entrata in Redazione il 31 luglio 1970.
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Schulenberger, J.R., Wilcox, C.H. Coerciveness inequalities for nonelliptic systems of partial differential equations. Annali di Matematica 88, 229–305 (1971). https://doi.org/10.1007/BF02415070
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DOI: https://doi.org/10.1007/BF02415070