Advertisement

On the strong unique continuation property of a degenerate elliptic operator with Hardy-type potential

  • Agnid BanerjeeEmail author
  • Arka Mallick
Article
  • 21 Downloads

Abstract

In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation
$$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$
(0.1)
where the potential V satisfies either of the following growth assumptions
$$\begin{aligned}&\left| V(z,t) \right| \le \frac{f(\rho (z,t))}{\rho (z,t)^2},\ \text {where} \rho \text { is as in }(2.1)\text { and }\nonumber \\&\quad f\text { satisfies the Dini integrability condition as in }(1.3). \end{aligned}$$
(0.2)
or when
$$\begin{aligned}&\left| V(z,t) \right| \le C \frac{\psi (z,t)^{\epsilon }}{\rho (z,t)^2},\ \text {for some }\epsilon >0\text { with }\psi \text { as in }(2.6)\text { and } N\text { even.} \end{aligned}$$
This extends some of the previous results obtained in [18] for this subfamily of Baouendi–Grushin operators. As corollaries, we obtain new unique continuation properties for solutions u to
$$\begin{aligned} \Delta _{{\mathbb {H}}} u = Vu \end{aligned}$$
with certain symmetries as expressed in (1.6) where \(\Delta _{{\mathbb {H}}}\) corresponds to the sub-Laplacian on the Heisenberg group \({\mathbb {H}}^n\).

Keywords

Unique continuation Baouendi–Grushin operator Hardy-type potential Carleman estimates 

Mathematics Subject Classification

35J70 35J75 

Notes

Acknowledgements

One of us, A.B., would like to thank Prof. Nicola Garofalo for several suggestions and feedbacks related to this work.

References

  1. 1.
    Amrein, W., Berthier, A., Georgescu, V.: \(L^{p}\) inequalities for the Laplacian and unique continuation. Ann. Inst. Fourier, Grenoble 31(3), 153–168 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aronszajn, N., Krzywicki, A., Szarski, J.: A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat. 4(1962), 417–453 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Almgren, F.: Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), pp. 1-6, North-Holland, Amsterdam-New York, (1979)Google Scholar
  4. 4.
    Banerjee, A.: Sharp vanishing order of solutions to stationary Schrodinger equations on Carnot groups of arbitrary step. J. Math. Anal. Appl. 465(1), 571–587 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banerjee, A., Garofalo, N.: Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators. Rend. Istit. Mat. Univ. Trieste 48, 189–207 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Baouendi, M.S.: Sur une classe d’operateurs elliptiques degeneres. (French). Bull. Soc. Math. France 95, 45–87 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bahouri, H.: Non prolongement unique des solutions d’opérateurs “somme de carrés” (French) [Failure of unique continuation for “sum of squares” operators]. Ann. Inst. Fourier (Grenoble) 36, 137–155 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Carleman, T.: Sur un probleme d’unicite pur les systemes d’equations aux derivees partielles a deux variables independantes. Ark. Mat. Astr. Fys. 26(17), 9 (1939)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chanillo, S., Sawyer, E.: Unique continuation for \(\Delta + \nu \) and the C. Fefferman-Phong class. Trans. Am. Math. Soc 318(1), 275–300 (1990)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Capogna, L., Danielli, D., Garofalo, N.: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Commun. PDE 18, 1765–1794 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Franchi, B., Gutierrez, C., Wheeden, R.: Two-weight Sobolev–Poincare inequalities and Harnack inequality for a class of degenerate elliptic operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 5(2), 167–175 (1994)Google Scholar
  13. 13.
    Franchi, B., Lanconelli, E.: Hölder regularity theorem for a class of linear nonuniformly elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10(4), 523–541 (1983)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Folland, G., Stein, E.: Estimates for \(\bar{\partial _b}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)CrossRefzbMATHGoogle Scholar
  15. 15.
    Escauriaza, L.: Carleman inequalities and the heat operator I. Duke Math. J. 104(1), 113–127 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Escauriaza, L., Vega, L.: Carleman inequalities and the heat operator II. Indiana Univ. Math. J 50(3), 1149–1169 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Escauriaza, L., Seregin, G., Sverak, V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169(2), 147–157 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Garofalo, N.: Unique continuation for a class of elliptic operators which degJ. Differential Equationsenerate on a manifold of arbitrary codimension. J. Differ. Equ. 104(1), 117–146 (1993)CrossRefzbMATHGoogle Scholar
  19. 19.
    Garofalo, N., Lin, F.: Monotonicity properties of variational integrals, \(A_p\) weights and unique continuation. Indiana Univ. Math. J. 35, 245–268 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Garofalo, N., Lin, F.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure Appl. Math. 40, 347–366 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Garofalo, N., Lanconelli, E.: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Ann. Inst. Fourier (Grenoble) 40, 313–356 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grušin, V.V.: A certain class of hypoelliptic operators. (Russian) Mat. Sb. (N.S.) 83(125), 456–473 (1970)Google Scholar
  23. 23.
    Grušin, V.V.: A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold. (Russian) Mat. Sb. (N.S.) 84(126), 163–195 (1971)Google Scholar
  24. 24.
    Garofalo, N., Rotz, Kevin: Properties of a frequency of Almgren type for harmonic functions in Carnot groups. Calc. Var. Partial Differ. Equ. 54(2), 2197–2238 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Garofalo, N., Shen, Z.: Carleman estimates for a subelliptic operator and unique continuation. Ann. Inst. Fourier (Grenoble) 44(1), 129–166 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Garofalo, N., Vassilev, D.: Strong unique continuation properties of generalized Baouendi-Grushin operators. Commun. Partial Differ. Equ. 32(4–6), 643–663 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Jerison, D.: Carleman inequalities for the Dirac and Laplace operators and unique continuation. Adv. Math. 62(2), 118–134 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jerison, D., Kenig, C.: Unique continuation and absence of positive eigenvalues for Schrodinger operators. Ann. Math. (2) 121(3), 463–494 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Koch, H., Petrosyan, A., Shi, W.: Higher regularity of the free boundary in the elliptic Signorini problem. Nonlinear Anal. 126, 3–44 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Koch, H., Tataru, D.: Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Commun. Pure Appl. Math. 54(3), 339–360 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pan, Y.: Unique continuation for Schrodinger operators with singular potentials. Commun. Partial Differ. Equ. 17(5–6), 953–965 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Poon, C.C.: Unique continuation for parabolic equations. Commun. Partial Differ. Equ. 21(3–4), 521–539 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Xu, C.J.: Subelliptic variational problems. Bull. Soc. Math. France 118, 147–169 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.TIFR Centre for Applicable MathematicsBangaloreIndia
  2. 2.Department of MathematicsEPFL SB CAMALausanneSwitzerland

Personalised recommendations