Nonzero eigenfunctions and A-harmonic functions of uniformly elliptic C1-coefficient operators, plus a complex-valued heat-equation solution, can achieve the maximal allowed double-exponential decay in cylinders.
Optimal unique continuation for periodic elliptic equations on large scales
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abstract
We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale on which they are valid, and are at least "almost" optimal in the prefactor constants--up to, at most, an iterated logarithm of the initial doubling ratio.
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Eigenfunctions with double exponential rate of localization
Nonzero eigenfunctions and A-harmonic functions of uniformly elliptic C1-coefficient operators, plus a complex-valued heat-equation solution, can achieve the maximal allowed double-exponential decay in cylinders.