Startseite Improved quantitative unique continuation for complex-valued drift equations in the plane
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Blair Davey , Carlos Kenig und Jenn-Nan Wang EMAIL logo
Veröffentlicht/Copyright: 29. Juni 2022
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 34 Heft 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.

References

[1] J. Bourgain and C. E. Kenig, On localization in the continuous Anderson–Bernoulli model in higher dimension, Invent. Math. 161 (2005), no. 2, 389–426. 10.1007/s00222-004-0435-7Suche in Google Scholar

[2] B. Davey, Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator, Comm. Partial Differential Equations 39 (2014), no. 5, 876–945. 10.1080/03605302.2013.796380Suche in Google Scholar

[3] B. Davey, On Landis’ conjecture in the plane for some equations with sign-changing potentials, Rev. Mat. Iberoam. 36 (2020), no. 5, 1571–1596. 10.4171/rmi/1176Suche in Google Scholar

[4] B. Davey, Quantitative unique continuation for Schrödinger operators, J. Funct. Anal. 279 (2020), no. 4, Article ID 108566. 10.1016/j.jfa.2020.108566Suche in Google Scholar

[5] B. Davey, C. Kenig and J.-N. Wang, The Landis conjecture for variable coefficient second-order elliptic PDEs, Trans. Amer. Math. Soc. 369 (2017), no. 11, 8209–8237. 10.1090/tran/7073Suche in Google Scholar

[6] B. Davey, C. Kenig and J.-N. Wang, On Landis’ conjecture in the plane when the potential has an exponentially decaying negative part, Algebra i Analiz 31 (2019), no. 2, 204–226. 10.1090/spmj/1600Suche in Google Scholar

[7] B. Davey, C.-L. Lin and J.-N. Wang, Strong unique continuation for the Lamé system with less regular coefficients, Math. Ann. 381 (2021), no. 1–2, 1005–1029. 10.1007/s00208-020-02026-0Suche in Google Scholar

[8] B. Davey and J.-N. Wang, Landis’ conjecture for general second order elliptic equations with singular lower order terms in the plane, J. Differential Equations 268 (2020), no. 3, 977–1042. 10.1016/j.jde.2019.08.035Suche in Google Scholar

[9] B. Davey and J. Zhu, Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions, Calc. Var. Partial Differential Equations 57 (2018), no. 3, Paper No. 92. 10.1007/s00526-018-1345-7Suche in Google Scholar

[10] B. Davey and J. Zhu, Quantitative uniqueness of solutions to second-order elliptic equations with singular lower order terms, Comm. Partial Differential Equations 44 (2019), no. 11, 1217–1251. 10.1080/03605302.2019.1629957Suche in Google Scholar

[11] C. Kenig, L. Silvestre and J.-N. Wang, On Landis’ conjecture in the plane, Comm. Partial Differential Equations 40 (2015), no. 4, 766–789. 10.1080/03605302.2014.978015Suche in Google Scholar

[12] C. Kenig and J.-N. Wang, Quantitative uniqueness estimates for second order elliptic equations with unbounded drift, Math. Res. Lett. 22 (2015), no. 4, 1159–1175. 10.4310/MRL.2015.v22.n4.a10Suche in Google Scholar

[13] V. A. Kondrat’ev and E. M. Landis, Qualitative theory of second-order linear partial differential equations, Partial Differential Equations. 3 (in Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Moscow (1988), 99–215, 220. 10.1007/978-3-642-58173-1_2Suche in Google Scholar

[14] C.-L. Lin and J.-N. Wang, Quantitative uniqueness estimates for the general second order elliptic equations, J. Funct. Anal. 266 (2014), no. 8, 5108–5125. 10.1016/j.jfa.2014.02.016Suche in Google Scholar

[15] A. Logunov, Nodal sets of Laplace eigenfunctions: Polynomial upper estimates of the Hausdorff measure, Ann. of Math. (2) 187 (2018), no. 1, 221–239. 10.4007/annals.2018.187.1.4Suche in Google Scholar

[16] A. Logunov, Nodal sets of Laplace eigenfunctions: Proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Ann. of Math. (2) 187 (2018), no. 1, 241–262. 10.4007/annals.2018.187.1.5Suche in Google Scholar

[17] A. Logunov and E. Malinnikova, Ratios of harmonic functions with the same zero set, Geom. Funct. Anal. 26 (2016), no. 3, 909–925. 10.1007/s00039-016-0369-4Suche in Google Scholar

[18] A. Logunov, E. Malinnikova, N. Nadirashvili and F. Nazarov, The Landis conjecture on exponential decay, preprint (2020), https://arxiv.org/abs/2007.07034. Suche in Google Scholar

[19] V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Mat. Sb. 182 (1991), no. 3, 364–383. Suche in Google Scholar
Received: 2022-04-12
Published Online: 2022-06-29
Published in Print: 2022-11-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Carleman estimates; elliptic systems; quantitative unique continuation

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 20.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2022-0114/html
Button zum nach oben scrollen