Summary
J. R. Schulenberger and C. H. Wilcox[1], [2], have proven a coerciveness inequality for a class of nonelliptic first-order partial differential operators of the form Λ = =E−1 AjDj, where Aj (j=1, ..., n) is a constant m × m Hermitian matrix, E=E(x) is uniformly positive definite, bounded, and uniformly differentiable Hermitian m × m matrix, and where the symbol Λ(p, x)=E(x)−1 Ajpj has constant rank for all p ε Rn − {0} and x ε Rn. They prove coerciveness on N(Λ)⊥, the orthogonal complement of the null space N(Λ) relative to the inner product
. Their proof is rather long, a simpler and shorter proof is given here. This proof leads naturally to a generalization of their results to the case where the Aj's need not be Hermitian.
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J. R. Schulenberger -C. H. Wilcox,Coerciveness inequalities for nonelliptic systems of partial differential equations, Ann. Mat. Pura Appl.,87 (to appear, 1971).
J. R. Schulenberger -C. H. Wilcox,A coerciveness inequality for a class of nonelliptic operators of constant deficit, Technical Summary Report no. 8, Department of Mathematics, University of Denver (October 1970).
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Entrato in Redazione il 31 marzo 1971.
Work supported in part by NSF grant GP-17526.
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Sarason, L. Remarks on an inequality of Schulenberger and Wilcox. Annali di Matematica 92, 23–28 (1972). https://doi.org/10.1007/BF02417933
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DOI: https://doi.org/10.1007/BF02417933