# Coerciveness inequalities for nonelliptic systems of partial differential equations

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## Abstract

This paper deals with first-order matrix partial differential operators of the form where x=(x for all u∈D(L)∩V, where D(L) denotes the domain of L and ∥·∥ denotes the norm in L

$$L = - i \mathop \sum \limits_{j = 1}^n L_j (x)D_i + L_o (x)$$

(1)

_{1}, ..., x_{n})∈R^{n}, D_{j}=∂/∂x_{j}, and the L_{j}(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 R^{n}. 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)

_{2, m}. L is said to have constant deficit k in R^{n}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∈R^{n}and p∈R^{n}−{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 A_{j}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 A_{j}, that*Λ*is coercive on the subspace N(Λ)^{⊥}, the orthogonal complement in H of N*(Λ)*, the nullspace of*Λ*.## Keywords

Differential Equation Hilbert Space Partial Differential Equation Linear Operator Differential Operator
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© Nicola Zanichelli Editore 1971