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Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces

  • Alexander Bukhgeim , Jin Cheng , Cristina C. Domme EMAIL logo and Shuai Lu
Published/Copyright: February 27, 2025
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 33 Issue 3

Abstract

We obtain new estimates of stability for the continuation of solutions of elliptic inequalities | ∂ ¯ ⁢ u | 2 ⩽ a ⁢ | u | 2 from finite discrete sets. For this, we use Hörmander’s type of Carleman estimates for ∂ ¯ with boundary terms and a special weight function equal to the linear combination of Green’s functions with singularities at points where our function is given. To estimate functions in a given domain we use not only the classical Blaschke partial sum S n but we also introduce a new sequence of numbers M n which gives us better estimates in the case of a finite number of observations.

Keywords: Carleman estimate; Blaschke sum; elliptic equation
MSC 2020: 35A05; 32W05; 30H20

Appendix

The figures below show the numbers M n and B n as functions of ρ for different values of r and n.

Figure 4 Graphing M n M_{n} and B n B_{n} for r = 0.1 r=0.1 and n = 5 n=5 .
Figure 4

Graphing M n and B n for r = 0.1 and n = 5 .

Figure 5 Graphing M n M_{n} and B n B_{n} for r = 0.1 r=0.1 and n = 10 n=10 .
Figure 5

Graphing M n and B n for r = 0.1 and n = 10 .

Figure 6 Graphing M n M_{n} and B n B_{n} for r = 0.1 r=0.1 and n = 50 n=50 .
Figure 6

Graphing M n and B n for r = 0.1 and n = 50 .

Figure 7 Graphing M n M_{n} and B n B_{n} for r = 0.1 r=0.1 and n = 100 n=100 .
Figure 7

Graphing M n and B n for r = 0.1 and n = 100 .

Figure 8 Graphing M n M_{n} and B n B_{n} for r = 0.5 r=0.5 and n = 5 n=5 .
Figure 8

Graphing M n and B n for r = 0.5 and n = 5 .

Figure 9 Graphing M n M_{n} and B n B_{n} for r = 0.5 r=0.5 and n = 10 n=10 .
Figure 9

Graphing M n and B n for r = 0.5 and n = 10 .

Figure 10 Graphing M n M_{n} and B n B_{n} for r = 0.5 r=0.5 and n = 50 n=50 .
Figure 10

Graphing M n and B n for r = 0.5 and n = 50 .

Figure 11 Graphing M n M_{n} and B n B_{n} for r = 0.5 r=0.5 and n = 100 n=100 .
Figure 11

Graphing M n and B n for r = 0.5 and n = 100 .

Figure 12 Graphing M n M_{n} and B n B_{n} for r = 0.9 r=0.9 and n = 5 n=5 .
Figure 12

Graphing M n and B n for r = 0.9 and n = 5 .

Figure 13 Graphing M n M_{n} and B n B_{n} for r = 0.9 r=0.9 and n = 10 n=10 .
Figure 13

Graphing M n and B n for r = 0.9 and n = 10 .

Figure 14 Graphing M n M_{n} and B n B_{n} for r = 0.9 r=0.9 and n = 50 n=50 .
Figure 14

Graphing M n and B n for r = 0.9 and n = 50 .

Figure 15 Graphing M n M_{n} and B n B_{n} for r = 0.9 r=0.9 and n = 100 n=100 .
Figure 15

Graphing M n and B n for r = 0.9 and n = 100 .

References

[1] A. L. Bukhgeĭm, Extension of solutions of elliptic equations from discrete sets, J. Inverse Ill-Posed Probl. 1 (1993), no. 1, 17–32. 10.1515/jiip.1993.1.1.17Search in Google Scholar

[2] T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys. 26 (1939), no. 17, 9. Search in Google Scholar

[3] J. B. Garnett, Bounded Analytic Functions, Pure Appl. Math. 96, Academic Press, New York, 1981. Search in Google Scholar

[4] L. Hörmander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand, Princeton, 1966. Search in Google Scholar

[5] M. M. Lavrent’ev, Some Improperly Posed Problems of Mathematical Physics, Springer Tracts Natural Philos. 11, Springer, Berlin, 1967. 10.1007/978-3-642-88210-4Search in Google Scholar

[6] S. P. Shishat’skiĭ, Interior problems for elliptic and parabolic equations, and Carleman estimates with distributed singularities, Sov. Math Dokl. 36 (1988), no. 3, 516–518. Search in Google Scholar
Received: 2025-01-27
Accepted: 2025-01-29
Published Online: 2025-02-27
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An inverse problem for non-selfadjoint Dirac operator with transmission conditions at finite interior points
  3. Inverse Sturm–Liouville problem with polynomials in the boundary condition and multiple eigenvalues
  4. Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
  5. On an inverse problem with high-order overdetermination conditions
  6. Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation
  7. Integrating the probe and singular sources methods: III. Mixed obstacle case
  8. Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time
  9. Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces
https://doi.org/10.1515/jiip-2025-0009
Keywords for this article
Carleman estimate; Blaschke sum; elliptic equation

Articles in the same Issue

  1. Frontmatter
  2. An inverse problem for non-selfadjoint Dirac operator with transmission conditions at finite interior points
  3. Inverse Sturm–Liouville problem with polynomials in the boundary condition and multiple eigenvalues
  4. Blow-up of solutions to some classes of ill-posed operator equations and a growth quantification result
  5. On an inverse problem with high-order overdetermination conditions
  6. Two regularization methods for identifying the source term of Caputo–Hadamard type time fractional diffusion-wave equation
  7. Integrating the probe and singular sources methods: III. Mixed obstacle case
  8. Stability estimate and the Tikhonov regularization method for the Kuramoto–Sivashinsky equation backward in time
  9. Continuation of solutions of elliptic systems from discrete sets with application to geometry of surfaces
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