Taiwanese Journal of Mathematics

Strict Monotonicity and Unique Continuation for General Non-local Eigenvalue Problems

Silvia Frassu and Antonio Iannizzotto

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Abstract

We consider the weighted eigenvalue problem for a general non-local pseudo-differential operator, depending on a bounded weight function. For such problem, we prove that strict (decreasing) monotonicity of the eigenvalues with respect to the weight function is equivalent to the unique continuation property of eigenfunctions. In addition, we discuss some unique continuation results for the special case of the fractional Laplacian.

Article information

Source
Taiwanese J. Math., Advance publication (2019), 14 pages.

Dates
First available in Project Euclid: 8 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1565229625

Digital Object Identifier
doi:10.11650/tjm/190709

Subjects
Primary: 35R11: Fractional partial differential equations 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx] 47A75: Eigenvalue problems [See also 47J10, 49R05]

Keywords
non-local operators eigenvalue problems unique continuation

Citation

Frassu, Silvia; Iannizzotto, Antonio. Strict Monotonicity and Unique Continuation for General Non-local Eigenvalue Problems. Taiwanese J. Math., advance publication, 8 August 2019. doi:10.11650/tjm/190709. https://projecteuclid.org/euclid.twjm/1565229625


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References

  • C. Anedda, F. Cuccu and S. Frassu, Minimization and Steiner symmetry of the first eigenvalue for a fractional eigenvalue problem with indefinite weight, arXiv:1904.02923.
  • M. M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations 39 (2014), no. 2, 354–397.
  • ––––, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5827–5867.
  • D. G. de Figueiredo, Positive solutions of semilinear elliptic problems, in: Differential Equations (S ao Paulo, 1981), 34–87, Lecture Notes in Mathematics 957, Springer, Berlin, 1982.
  • D. G. de Figueiredo and J.-P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), no. 1-2, 339–346.
  • S. Frassu, Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$, Commun. Pure Appl. Anal. 18 (2019), no. 4, 1847–1867.
  • T. Ghosh, A. Rüland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, arXiv:1801.04449.
  • A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006.
  • A. Iannizzotto and N. S. Papageorgiou, Existence and multiplicity results for resonant fractional boundary value problems, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 511–532.
  • O. Martio, Counterexamples for unique continuation, Manuscripta Math. 60 (1988), no. 1, 21–47.
  • G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications 162, Cambridge University Press, Cambridge, 2016.
  • K. Perera, R. P. Agarwal and D. O'Regan, Morse Theoretic Aspects of $p$-Laplacian Type Operators, Mathematical Surveys and Monographs 161, American Mathematical Society, Providence, RI, 2010.
  • X. Ros-Oton, Nonlocal elliptic equations in bounded domains: a survey, Publ. Mat. 60 (2016), no. 1, 3–26.
  • A. Rüland, On Some Rigidity Properties in PDEs, dissertation, University of Bonn, 2014.
  • ––––, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations 40 (2015), no. 1, 77–114.
  • I. Seo, On unique continuation for Schrödinger operators of fractional and higher orders, Math. Nachr. 287 (2014), no. 5-6, 699–703.
  • ––––, Carleman inequalities for fractional Laplacians and unique continuation, Taiwanese J. Math. 19 (2015), no. 5, 1533–1540.
  • R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137.
  • ––––, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 4, 831–855.
  • T. H. Wolff, Note on counterexamples in strong unique continuation problems, Proc. Amer. Math. Soc. 114 (1992), no. 2, 351–356.
  • H. Yu, Unique continuation for fractional orders of elliptic equations, Ann. PDE 3 (2017), no. 2, Art. 16, 21 pp.