Annali di Matematica Pura ed Applicata

, Volume 88, Issue 1, pp 229–305 | Cite as

Coerciveness inequalities for nonelliptic systems of partial differential equations

  • John R. Schulenberger
  • Calvin H. Wilcox


This paper deals with first-order matrix partial differential operators of the form
$$L = - i \mathop \sum \limits_{j = 1}^n L_j (x)D_i + L_o (x)$$
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
$$\mathop \sum \limits_{j = 1}^n \left\| {D_j u} \right\|^2 \leqslant \mu ^2 (\left\| {Lu} \right\|^2 + \left\| u \right\|^2 )$$
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.


Differential Equation Hilbert Space Partial Differential Equation Linear Operator Differential Operator 
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Copyright information

© Nicola Zanichelli Editore 1971

Authors and Affiliations

  • John R. Schulenberger
  • Calvin H. Wilcox

There are no affiliations available

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