{'fraction_non_alphanumeric': 0.019672131147540985, 'fraction_numerical': 0.001639344262295082, 'mean_word_length': 5.0495049504950495, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Universal inequalities for eigenvalues of a system of elliptic equations

Let D be a bounded domain in an n-dimensional Euclidean space ℝn. Assume that are eigenvalues of an eigenvalue problem of a system of n elliptic equations: In particular, when n=3, the eigenvalue problem describes the behaviour of the elastic vibration. We obtain universal inequalities for eigenvalues of the above eigenvalue problem by making use of a direct and explicit method; our results are sharper than one of Hook. Furthermore, a universal inequality for lower-order eigenvalues of the above eigenvalue problem is also derived.