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November  2021, 41(11): 5105-5139. doi: 10.3934/dcds.2021070

Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation

1. 

Tata Institute of Fundamental Research, Centre For Applicable Mathematics, Bangalore-560065, India

2. 

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Received  November 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Fund Project: A.B is supported in part by SERB Matrix grant MTR/2018/000267 and by Department of Atomic Energy, Government of India, under project no. 12-R & D-TFR-5.01-0520.
R.M is supported by [DST/INSPIRE/04/2019/00914]

In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (ⅰ) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ⅱ) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

Citation: Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5105-5139. doi: 10.3934/dcds.2021070
References:
[1]

F. J. Almgren, Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal Submanifolds and Geodesics, North-Holland, Amsterdam-New York, (1979), 1–6.  Google Scholar

[2]

N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249.   Google Scholar

[3]

N. AronszajnA. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453.  doi: 10.1007/BF02591624.  Google Scholar

[4]

V. Arya and A. Banerjee, Strong backward uniqueness for sublinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 18 pp. doi: 10.1007/s00030-020-00657-5.  Google Scholar

[5]

L. Bakri, Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Comm. Partial Differential Equations, 38 (2013), 69-91.  doi: 10.1080/03605302.2012.736912.  Google Scholar

[6]

L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580.  doi: 10.1512/iumj.2012.61.4713.  Google Scholar

[7]

A. Banerjee, Sharp vanishing order of solutions to stationary Schrödinger equations on Carnot groups of arbitrary step, J. Math. Anal. Appl., 465 (2018), 571-587.  doi: 10.1016/j.jmaa.2018.05.029.  Google Scholar

[8]

A. Banerjee and N. Garofalo, Quantitative uniqueness for elliptic equations at the boundary of $C^{1, Dini}$ domains, J. Differential Equations, 261 (2016), 6718-6757.  doi: 10.1016/j.jde.2016.09.001.  Google Scholar

[9]

A. Banerjee and N. Garofalo, Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 189-207.  doi: 10.13137/2464-8728/13156.  Google Scholar

[10]

A. Banerjee, N. Garofalo and R. Manna, Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation, Applicable Analysis, (2019), arXiv: 1903.08382. doi: 10.1080/00036811.2020.1713314.  Google Scholar

[11]

A. Banerjee and A. Mallick, On the strong unique continuation of a degenerate elliptic operator with Hardy type potential, Ann. Mat. Pura Appl., 199 (2020), 1-21.  doi: 10.1007/s10231-019-00864-7.  Google Scholar

[12]

A. Banerjee and R. Manna, Space like strong unique continuation for sublinear parabolic equations, J. Lond. Math. Soc. (2), 102 (2020), 205-228.  doi: 10.1112/jlms.12317.  Google Scholar

[13]

S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87.  Google Scholar

[14]

J. Bourgain and C. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426.  doi: 10.1007/s00222-004-0435-7.  Google Scholar

[15]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[16]

L. A. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[17]

T. Carleman, Sur un problème d'unicité pur les systemes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., 26 (1939), 9 pp.  Google Scholar

[18]

S. Chanillo and E. Sawyer, Unique continuation for $\Delta + \nu$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.  doi: 10.2307/2001239.  Google Scholar

[19]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.  doi: 10.1007/BF01393691.  Google Scholar

[20]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262.  Google Scholar

[21]

L. Escauriaza and S. Vessella, Optimal three-cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, Contemp. Math., 333 (2003), 79-87.  doi: 10.1090/conm/333/05955.  Google Scholar

[22]

M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.  Google Scholar

[23]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.  Google Scholar

[24]

B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégén'er'es, Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, (1984), 105–114.  Google Scholar

[25]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, 4 (1983), 523–541.  Google Scholar

[26]

B. Franchi and E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations, 9 (1984), 1237-1264.  doi: 10.1080/03605308408820362.  Google Scholar

[27]

N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146.  doi: 10.1006/jdeq.1993.1065.  Google Scholar

[28]

N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble), 40 (1990), 313-356.  doi: 10.5802/aif.1215.  Google Scholar

[29]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245–268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[30]

N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.  doi: 10.1002/cpa.3160400305.  Google Scholar

[31]

N. Garofalo and X. Ros-Oton, Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam., 35 (2019), 1309-1365.  doi: 10.4171/rmi/1087.  Google Scholar

[32]

N. Garofalo and K. Rotz, Properties of a frequency of Almgren type for harmonic functions in Carnot groups, Calc. Var. Partial Differential Equations, 54 (2015), 2197-2238.  doi: 10.1007/s00526-015-0862-x.  Google Scholar

[33]

N. Garofalo and Z. Shen, Carleman estimates for a subelliptic operator and unique continuation, Ann. Inst. Fourier (Grenoble), 44 (1994), 129-166.  doi: 10.5802/aif.1392.  Google Scholar

[34]

N. Garofalo and D. Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations, 32 (2007), 643-663.  doi: 10.1080/03605300500532905.  Google Scholar

[35]

V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.   Google Scholar

[36]

V. V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.), 84 (1971), 163-195.   Google Scholar

[37]

L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64.  doi: 10.1080/03605308308820262.  Google Scholar

[38]

D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 62 (1986), 118-134.  doi: 10.1016/0001-8708(86)90096-4.  Google Scholar

[39]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2), 121 (1985), 463-494.  doi: 10.2307/1971205.  Google Scholar

[40]

H. Koch, A. Petrosyan and W. Shi, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal., 126 (2015), 3–44. doi: 10.1016/j.na.2015.01.007.  Google Scholar

[41]

H. KochA. Rüland and W. Shi, The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.   Google Scholar

[42]

H. Koch and D. Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001), 339-360.  doi: 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D.  Google Scholar

[43]

V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Math. USSR-Sb., 72 (1992), 343-361.  doi: 10.1070/SM1992v072n02ABEH001414.  Google Scholar

[44]

K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal., 54 (1974), 105–117. doi: 10.1007/BF00247634.  Google Scholar

[45]

A. Plis, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 11 (1963), 95-100.   Google Scholar

[46]

R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217. doi: 10.1006/jdeq.1997.3327.  Google Scholar

[47]

A. Rüland, Unique Continuation for sublinear elliptic equations based on Carleman estimates, J. Differential Equations, 265 (2018), 6009-6035.  doi: 10.1016/j.jde.2018.07.025.  Google Scholar

[48]

A. Rüland, On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates, Trans. Amer. Math. Soc., 369 (2017), 2311-2362.  doi: 10.1090/tran/6758.  Google Scholar

[49]

Y. SireS. Terracini and G. Tortone, On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$ harmonic functions, J. Math. Pures Appl. (9), 143 (2020), 376-441.  doi: 10.1016/j.matpur.2020.01.010.  Google Scholar

[50]

N. Soave and T. Weth, The unique continuation property of sublinear equations, SIAM J. Math. Anal., 50 (2018), 3919–3938. doi: 10.1137/17M1144325.  Google Scholar

[51]

N. Soave and S. Terracini, The nodal set of solutions to some elliptic problems: Sublinear equations, and unstable two-phase membrane problem, Adv. Math., 334 (2018), 243-299.  doi: 10.1016/j.aim.2018.06.007.  Google Scholar

[52]

C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.  doi: 10.1215/S0012-7094-86-05303-2.  Google Scholar

[53]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[54]

G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652. Google Scholar

[55]

L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242.  doi: 10.1016/S0022-1236(03)00093-4.  Google Scholar

[56]

J. Zhu, Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762.  doi: 10.1353/ajm.2016.0027.  Google Scholar

show all references

References:
[1]

F. J. Almgren, Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal Submanifolds and Geodesics, North-Holland, Amsterdam-New York, (1979), 1–6.  Google Scholar

[2]

N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl., 36 (1957), 235-249.   Google Scholar

[3]

N. AronszajnA. Krzywicki and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., 4 (1962), 417-453.  doi: 10.1007/BF02591624.  Google Scholar

[4]

V. Arya and A. Banerjee, Strong backward uniqueness for sublinear parabolic equations, NoDEA Nonlinear Differential Equations Appl., 27 (2020), 18 pp. doi: 10.1007/s00030-020-00657-5.  Google Scholar

[5]

L. Bakri, Carleman estimates for the Schrödinger operator. Applications to quantitative uniqueness, Comm. Partial Differential Equations, 38 (2013), 69-91.  doi: 10.1080/03605302.2012.736912.  Google Scholar

[6]

L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580.  doi: 10.1512/iumj.2012.61.4713.  Google Scholar

[7]

A. Banerjee, Sharp vanishing order of solutions to stationary Schrödinger equations on Carnot groups of arbitrary step, J. Math. Anal. Appl., 465 (2018), 571-587.  doi: 10.1016/j.jmaa.2018.05.029.  Google Scholar

[8]

A. Banerjee and N. Garofalo, Quantitative uniqueness for elliptic equations at the boundary of $C^{1, Dini}$ domains, J. Differential Equations, 261 (2016), 6718-6757.  doi: 10.1016/j.jde.2016.09.001.  Google Scholar

[9]

A. Banerjee and N. Garofalo, Quantitative uniqueness for zero-order perturbations of generalized Baouendi-Grushin operators, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 189-207.  doi: 10.13137/2464-8728/13156.  Google Scholar

[10]

A. Banerjee, N. Garofalo and R. Manna, Carleman estimates for Baouendi-Grushin operators with applications to quantitative uniqueness and strong unique continuation, Applicable Analysis, (2019), arXiv: 1903.08382. doi: 10.1080/00036811.2020.1713314.  Google Scholar

[11]

A. Banerjee and A. Mallick, On the strong unique continuation of a degenerate elliptic operator with Hardy type potential, Ann. Mat. Pura Appl., 199 (2020), 1-21.  doi: 10.1007/s10231-019-00864-7.  Google Scholar

[12]

A. Banerjee and R. Manna, Space like strong unique continuation for sublinear parabolic equations, J. Lond. Math. Soc. (2), 102 (2020), 205-228.  doi: 10.1112/jlms.12317.  Google Scholar

[13]

S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87.  Google Scholar

[14]

J. Bourgain and C. Kenig, On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161 (2005), 389-426.  doi: 10.1007/s00222-004-0435-7.  Google Scholar

[15]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.  Google Scholar

[16]

L. A. CaffarelliS. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.  doi: 10.1007/s00222-007-0086-6.  Google Scholar

[17]

T. Carleman, Sur un problème d'unicité pur les systemes d'équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys., 26 (1939), 9 pp.  Google Scholar

[18]

S. Chanillo and E. Sawyer, Unique continuation for $\Delta + \nu$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc., 318 (1990), 275-300.  doi: 10.2307/2001239.  Google Scholar

[19]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183.  doi: 10.1007/BF01393691.  Google Scholar

[20]

H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262.  Google Scholar

[21]

L. Escauriaza and S. Vessella, Optimal three-cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, Contemp. Math., 333 (2003), 79-87.  doi: 10.1090/conm/333/05955.  Google Scholar

[22]

M. Fall and V. Felli, Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Comm. Partial Differential Equations, 39 (2014), 354-397.  doi: 10.1080/03605302.2013.825918.  Google Scholar

[23]

M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.  Google Scholar

[24]

B. Franchi and E. Lanconelli, Une métrique associée à une classe d'opérateurs elliptiques dégén'er'es, Rend. Sem. Mat. Univ. Politec. Torino 1983, Special Issue, (1984), 105–114.  Google Scholar

[25]

B. Franchi and E. Lanconelli, Hölder regularity theorem for a class of linear non uniformly elliptic operators with measurable coefficients, Ann. Sc. Norm. Sup. Pisa, 4 (1983), 523–541.  Google Scholar

[26]

B. Franchi and E. Lanconelli, An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality, Comm. Partial Differential Equations, 9 (1984), 1237-1264.  doi: 10.1080/03605308408820362.  Google Scholar

[27]

N. Garofalo, Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension, J. Diff. Equations, 104 (1993), 117-146.  doi: 10.1006/jdeq.1993.1065.  Google Scholar

[28]

N. Garofalo and E. Lanconelli, Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation, Ann. Inst. Fourier (Grenoble), 40 (1990), 313-356.  doi: 10.5802/aif.1215.  Google Scholar

[29]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245–268. doi: 10.1512/iumj.1986.35.35015.  Google Scholar

[30]

N. Garofalo and F.-H. Lin, Unique continuation for elliptic operators: A geometric-variational approach, Comm. Pure Appl. Math., 40 (1987), 347-366.  doi: 10.1002/cpa.3160400305.  Google Scholar

[31]

N. Garofalo and X. Ros-Oton, Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian, Rev. Mat. Iberoam., 35 (2019), 1309-1365.  doi: 10.4171/rmi/1087.  Google Scholar

[32]

N. Garofalo and K. Rotz, Properties of a frequency of Almgren type for harmonic functions in Carnot groups, Calc. Var. Partial Differential Equations, 54 (2015), 2197-2238.  doi: 10.1007/s00526-015-0862-x.  Google Scholar

[33]

N. Garofalo and Z. Shen, Carleman estimates for a subelliptic operator and unique continuation, Ann. Inst. Fourier (Grenoble), 44 (1994), 129-166.  doi: 10.5802/aif.1392.  Google Scholar

[34]

N. Garofalo and D. Vassilev, Strong unique continuation properties of generalized Baouendi-Grushin operators, Comm. Partial Differential Equations, 32 (2007), 643-663.  doi: 10.1080/03605300500532905.  Google Scholar

[35]

V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.   Google Scholar

[36]

V. V. Grushin, A certain class of elliptic pseudodifferential operators that are degenerate on a submanifold, Mat. Sb. (N.S.), 84 (1971), 163-195.   Google Scholar

[37]

L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64.  doi: 10.1080/03605308308820262.  Google Scholar

[38]

D. Jerison, Carleman inequalities for the Dirac and Laplace operators and unique continuation, Adv. in Math., 62 (1986), 118-134.  doi: 10.1016/0001-8708(86)90096-4.  Google Scholar

[39]

D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators, Ann. of Math. (2), 121 (1985), 463-494.  doi: 10.2307/1971205.  Google Scholar

[40]

H. Koch, A. Petrosyan and W. Shi, Higher regularity of the free boundary in the elliptic Signorini problem, Nonlinear Anal., 126 (2015), 3–44. doi: 10.1016/j.na.2015.01.007.  Google Scholar

[41]

H. KochA. Rüland and W. Shi, The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.   Google Scholar

[42]

H. Koch and D. Tataru, Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients, Comm. Pure Appl. Math., 54 (2001), 339-360.  doi: 10.1002/1097-0312(200103)54:3<339::AID-CPA3>3.0.CO;2-D.  Google Scholar

[43]

V. Z. Meshkov, On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Math. USSR-Sb., 72 (1992), 343-361.  doi: 10.1070/SM1992v072n02ABEH001414.  Google Scholar

[44]

K. Miller, Nonunique continuation for uniformly parabolic and elliptic equations in self-adjoint divergence form with Hölder continuous coefficients, Arch. Rational Mech. Anal., 54 (1974), 105–117. doi: 10.1007/BF00247634.  Google Scholar

[45]

A. Plis, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 11 (1963), 95-100.   Google Scholar

[46]

R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217. doi: 10.1006/jdeq.1997.3327.  Google Scholar

[47]

A. Rüland, Unique Continuation for sublinear elliptic equations based on Carleman estimates, J. Differential Equations, 265 (2018), 6009-6035.  doi: 10.1016/j.jde.2018.07.025.  Google Scholar

[48]

A. Rüland, On quantitative unique continuation properties of fractional Schrödinger equations: Doubling, vanishing order and nodal domain estimates, Trans. Amer. Math. Soc., 369 (2017), 2311-2362.  doi: 10.1090/tran/6758.  Google Scholar

[49]

Y. SireS. Terracini and G. Tortone, On the nodal set of solutions to degenerate or singular elliptic equations with an application to $s-$ harmonic functions, J. Math. Pures Appl. (9), 143 (2020), 376-441.  doi: 10.1016/j.matpur.2020.01.010.  Google Scholar

[50]

N. Soave and T. Weth, The unique continuation property of sublinear equations, SIAM J. Math. Anal., 50 (2018), 3919–3938. doi: 10.1137/17M1144325.  Google Scholar

[51]

N. Soave and S. Terracini, The nodal set of solutions to some elliptic problems: Sublinear equations, and unstable two-phase membrane problem, Adv. Math., 334 (2018), 243-299.  doi: 10.1016/j.aim.2018.06.007.  Google Scholar

[52]

C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65.  doi: 10.1215/S0012-7094-86-05303-2.  Google Scholar

[53]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[54]

G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652. Google Scholar

[55]

L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242.  doi: 10.1016/S0022-1236(03)00093-4.  Google Scholar

[56]

J. Zhu, Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762.  doi: 10.1353/ajm.2016.0027.  Google Scholar

[1]

Ihyeok Seo. Carleman estimates for the Schrödinger operator and applications to unique continuation. Communications on Pure &amp; Applied Analysis, 2012, 11 (3) : 1013-1036. doi: 10.3934/cpaa.2012.11.1013

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