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Recently, the eigenvalue problems formulated with symmetric positive definitebilinear forms have been well investigated with the aim of explicit bounds forthe eigenvalues. In this paper, the existing theorems for bounding eigenvaluesare further extended to deal with the case of eigenvalue problems defined bypositive semi-definite bilinear forms. As an application, the eigenvalueestimation theorems are applied to the error constant estimation for polynomialprojections.

Explicit eigenvalue bounds of differential operators defined by symmetric positive semi-definite bilinear forms