Gaps in the essential spectrum of periodic elastic waveguides

Examples of periodic elastic waveguides are constructed, the essential spectrum of which has a gap, i.e. an open interval in the positive real semiaxis intersecting with the discrete spectrum only. The gap is detected with the help of an inequality of Korn's type and the max-min principle for eigenvalues of self-adjoint positive operators. Under a certain symmetry assumption, it is demonstrated that the first band of the essential spectrum can include eigenvalues in the point spectrum.

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