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On quantitative uniqueness for elliptic equations

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Abstract

We address the question of quantitative uniqueness for the equation \(\Delta u=V u\) with either periodic or Dirichlet boundary conditions in a disk. We construct solutions u corresponding to potential functions V such that u vanishes of order \(\mathrm{const} \Vert V\Vert _{L^\infty }^{2/3}\). The example also shows sharpness of recently obtained bounds in the case of a parabolic equation \(u_t-\Delta u= V u\).

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Acknowledgements

The authors were supported in part by the NSF Grant DMS-1615239.

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Guher Camliyurt & Igor Kukavica

  2. Department of Mathematics, University of Maryland, College Park, MD, 20740, USA

    Fei Wang

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  1. Guher Camliyurt
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  2. Igor Kukavica
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  3. Fei Wang
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Correspondence to Igor Kukavica.

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Camliyurt, G., Kukavica, I. & Wang, F. On quantitative uniqueness for elliptic equations. Math. Z. 291, 227–244 (2019). https://doi.org/10.1007/s00209-018-2081-6

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