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

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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

Agnid Banerjee ^1, and Ramesh Manna ^2,

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.

Keywords: Grushin, Carleman estimates, unique continuation, quantitative uniqueness.

Mathematics Subject Classification: 35A23, 35B40, 35B45, 35B60.

Citation: Agnid Banerjee, Ramesh Manna. Carleman estimates for a class of variable coefficient degenerate elliptic operators with applications to unique continuation. Discrete and 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.
[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.
[3]	N. Aronszajn, A. 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.
[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.
[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.
[6]	L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580. doi: 10.1512/iumj.2012.61.4713.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87.
[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.
[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.
[16]	L. A. Caffarelli, S. 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.
[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.
[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.
[19]	H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183. doi: 10.1007/BF01393691.
[20]	H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
[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.
[37]	L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64. doi: 10.1080/03605308308820262.
[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.
[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.
[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.
[41]	H. Koch, A. Rüland and W. Shi, The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.
[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.
[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.
[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.
[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.
[46]	R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217. doi: 10.1006/jdeq.1997.3327.
[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.
[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.
[49]	Y. Sire, S. 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.
[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.
[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.
[52]	C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. doi: 10.1215/S0012-7094-86-05303-2.
[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.
[54]	G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652.
[55]	L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242. doi: 10.1016/S0022-1236(03)00093-4.
[56]	J. Zhu, Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762. doi: 10.1353/ajm.2016.0027.

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.
[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.
[3]	N. Aronszajn, A. 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.
[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.
[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.
[6]	L. Bakri, Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61 (2012), 1565-1580. doi: 10.1512/iumj.2012.61.4713.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	S. M. Baouendi, Sur une classe d'opérateurs elliptiques dégénérés, Bull. Soc. Math. France, 95 (1967), 45–87.
[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.
[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.
[16]	L. A. Caffarelli, S. 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.
[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.
[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.
[19]	H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93 (1988), 161-183. doi: 10.1007/BF01393691.
[20]	H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, Analysis, Et Cetera, Academic Press, Boston, MA, (1990), 251–262.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[35]	V. V. Grushin, A certain class of hypoelliptic operators, Mat. Sb. (N.S.), 83 (1970), 456-473.
[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.
[37]	L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations, 8 (1983), 21-64. doi: 10.1080/03605308308820262.
[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.
[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.
[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.
[41]	H. Koch, A. Rüland and W. Shi, The variable coefficient thin obstacle problem: Higher regularity, Adv. Differential Equations, 22 (2017), 793-866.
[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.
[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.
[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.
[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.
[46]	R. Regbaoui, Strong uniqueness for second order differential operators, J. Differential Equations, 141 (1997), 201–217. doi: 10.1006/jdeq.1997.3327.
[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.
[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.
[49]	Y. Sire, S. 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.
[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.
[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.
[52]	C. D. Sogge, Oscillatory integrals and spherical harmonics, Duke Math. J., 53 (1986), 43-65. doi: 10.1215/S0012-7094-86-05303-2.
[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.
[54]	G. Tortone, The nodal set of solutions to some nonlocal sublinear problems, arXiv: 2004.04652.
[55]	L. Wang, Hölder estimates for subelliptic operators, J. Funct. Anal., 199 (2003), 228-242. doi: 10.1016/S0022-1236(03)00093-4.
[56]	J. Zhu, Quantitative uniqueness for elliptic equations, Amer. J. Math., 138 (2016), 733-762. doi: 10.1353/ajm.2016.0027.

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