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Improved quantitative unique continuation for complex-valued drift equations in the plane

  • Blair Davey , Carlos Kenig and Jenn-Nan Wang EMAIL logo
Published/Copyright: June 29, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 6

Abstract

In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane. We consider equations of the form Δ ⁢ u + W ⋅ ∇ ⁡ u = 0 in ℝ 2 , where W = W 1 + i ⁢ W 2 with each W j being real-valued. Under the assumptions that W j ∈ L q j for some q 1 ∈ [ 2 , ∞ ] , q 2 ∈ ( 2 , ∞ ] and that W 2 exhibits rapid decay at infinity, we prove new global unique continuation estimates. This improvement is accomplished by reducing our equations to vector-valued Beltrami systems. Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme.

Keywords: Carleman estimates; elliptic systems; quantitative unique continuation
MSC 2010: 35J47; 35J10; 35J05

Communicated by Christopher D. Sogge

Funding source: National Science Foundation

Award Identifier / Grant number: DMS-2137743

Award Identifier / Grant number: DMS-1800082

Funding source: Simons Foundation

Award Identifier / Grant number: 430198

Funding statement: Part of this research was carried out while the first author was visiting the National Center for Theoretical Sciences (NCTS) at National Taiwan University. The first author wishes to the thank the NCTS for their financial support and their kind hospitality during her visit to Taiwan. The first author is supported in part by NSF DMS-2137743 and Simons Foundation Collaboration Grant 430198. The second author is supported in part by NSF DMS-1800082. The third author is supported in part by MOST 108-2115-M-002-002-MY3 and MOST 109-2115-M-002-001-MY3.

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Received: 2022-04-12
Published Online: 2022-06-29
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
https://doi.org/10.1515/forum-2022-0114
Keywords for this article
Carleman estimates; elliptic systems; quantitative unique continuation

Articles in the same Issue

  1. Frontmatter
  2. Wong–Zakai approximations and support theorems for stochastic McKean–Vlasov equations
  3. The Singer transfer for infinite real projective space
  4. The Stiefel–Whitney classes of moment-angle manifolds are trivial
  5. On almost p-rational characters of p ′ -degree
  6. Homogeneous four-manifolds with half-harmonic Weyl curvature tensor
  7. Construction of a class of spectral measures
  8. On the number of prime divisors and radicals of non-zero Fourier coefficients of Hilbert cusp forms
  9. Molecular characterization of weak Hardy spaces associated with ball quasi-Banach function spaces on spaces of homogeneous type with its applications to Littlewood–Paley function characterizations
  10. Conway invariant Jacobi forms on the Leech lattice
  11. Geometric characterizations of Gromov hyperbolic Hölder domains
  12. Improved quantitative unique continuation for complex-valued drift equations in the plane
  13. On uniqueness of branching to fixed point Lie subalgebras
  14. Fefferman type criterion on weighted bi-parameter local Hardy spaces and boundedness of bi-parameter pseudodifferential operators
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