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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
Article

Abstract

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

Keywords

Differential Equation Hilbert Space Partial Differential Equation Linear Operator Differential Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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