Space-like quantitative uniqueness for parabolic operators. (English. French summary) Zbl 1521.35005
Summary: We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a \(C^1\) potential \(V\). Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.
MSC:
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B60 | Continuation and prolongation of solutions to PDEs |
| 35K10 | Second-order parabolic equations |
References:
| [1] | Arya, V.; Banerjee, A., Quantitative uniqueness for fractional heat type operators · Zbl 1518.35008 |
| [2] | Bakri, L., Quantitative uniqueness for Schrödinger operator, Indiana Univ. Math. J., 61, 4, 1565-1580 (2012) · Zbl 1279.58011 |
| [3] | Bakri, L.; Casteras, J., Quantitative uniqueness for Schrödinger operator with regular potentials, Math. Methods Appl. Sci., 37, 1992-2008 (2014) · Zbl 1301.35190 |
| [4] | Banerjee, A.; Garofalo, N., Quantitative uniqueness for elliptic equations at the boundary of \(C^{1 , D i n i}\) domains, J. Differ. Equ., 261, 12, 6718-6757 (2016) · Zbl 1359.35039 |
| [5] | Bourgain, J.; Kenig, C., On localization in the continuous Anderson-Bernoulli model in higher dimension, Invent. Math., 161, 389-426 (2005) · Zbl 1084.82005 |
| [6] | Banerjee, A.; Krishnan, V.; Senapati, S., The Calderön problem for space-time fractional parabolic operators with variable coefficients · Zbl 07879906 |
| [7] | Bellova, K.; Lin, F., Nodal sets of Steklov eigenfunctions, Calc. Var. Partial Differ. Equ., 54, 2, 2239-2268 (2015) · Zbl 1327.35263 |
| [8] | Camliyurt, G.; Kukavica, I., Quantitative unique continuation for a parabolic equation, Indiana Univ. Math. J., 67, 2, 657-678 (2018) · Zbl 1402.35134 |
| [9] | Donnelly, H.; Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93, 161-183 (1988) · Zbl 0659.58047 |
| [10] | Donnelly, H.; Fefferman, C., Nodal sets of eigenfunctions: Riemannian manifolds with boundary, (Analysis, Et Cetera (1990), Academic Press: Academic Press Boston, MA), 251-262 · Zbl 0697.58055 |
| [11] | Davey, B.; Kenig, C.; Wang, J. N., The Landis conjecture for variable coefficient second-order elliptic PDEs, Trans. Am. Math. Soc., 369, 11, 8209-8237 (2017) · Zbl 1375.35061 |
| [12] | Escauriaza, L.; Fernandez, F., Unique continuation for parabolic operators, Ark. Mat., 41, 1, 35-60 (2003), (English summary) · Zbl 1028.35052 |
| [13] | Escauriaza, L.; Fernandez, F.; Vessella, S., Doubling properties of caloric functions, Appl. Anal., 85, 1-3, 205-223 (2006) · Zbl 1090.35050 |
| [14] | Escauriaza, L.; Kenig, C.; Ponce, G.; Vega, L., Hardy’s uncertainty principle, convexity and Schrödinger evolutions, J. Eur. Math. Soc., 10, 4, 883-907 (2008) · Zbl 1158.35018 |
| [15] | Escauriaza, L.; Seregin, G.; Sverak, V., Backward uniqueness for the heat operator in a half-space, St. Petersburg Math. J., 15, 139-148 (2004) · Zbl 1053.35052 |
| [16] | Escauriaza, L.; Vessella, S., Optimal three cylinder inequalities for solutions to parabolic equations with Lipschitz leading coefficients, (Inverse Problems: Theory and Applications. Inverse Problems: Theory and Applications, Cortona/Pisa, 2002. Inverse Problems: Theory and Applications. Inverse Problems: Theory and Applications, Cortona/Pisa, 2002, Contemp. Math., vol. 333 (2003), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 79-87 · Zbl 1056.35150 |
| [17] | Friedman, A., Partial Differential Equations of Parabolic Type (1964), Prentice-Hall, Inc.: Prentice-Hall, Inc. Englewood Cliffs, xiv+347 pp. · Zbl 0144.34903 |
| [18] | Garofalo, N.; Lin, F., Monotonicity properties of variational integrals, \( A_p\) weights and unique continuation, Indiana Univ. Math. J., 35, 245-268 (1986) · Zbl 0678.35015 |
| [19] | Garofalo, N.; Lin, F., Unique continuation for elliptic operators: a geometric-variational approach, Commun. Pure Appl. Math., 40, 347-366 (1987) · Zbl 0674.35007 |
| [20] | Kukavica, I., Quantitative uniqueness for second order elliptic operators, Duke Math. J., 91, 225-240 (1998) · Zbl 0947.35045 |
| [21] | Kukavica, I.; Le, Q., On quantitative uniqueness for parabolic equations, J. Differ. Equ., 341, 438-480 (2022) · Zbl 1504.35003 |
| [22] | Kenig, C.; Silvestre, L.; Wang, J. N., On Landis’ conjecture in the plane, Commun. Partial Differ. Equ., 40, 4, 766-789 (2015) · Zbl 1320.35119 |
| [23] | Koch, H.; Tataru, D., Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients, Commun. Partial Differ. Equ., 34, 4-6, 305-366 (2009) · Zbl 1178.35107 |
| [24] | Lieberman, G., Second Order Parabolic Differential Equations (1996), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ, xii+439 pp. · Zbl 0884.35001 |
| [25] | Logunov, A., Nodal sets of Laplace eigenfunctions: proof of Nadirashvili’s conjecture and of the lower bound in Yau’s conjecture, Ann. Math., 187, 241-262 (2018) · Zbl 1384.58021 |
| [26] | Logunov, A., Nodal sets of Laplace eigenfunctions: polynomial upper estimates of the Hausdorff measure, Ann. Math., 187, 221-239 (2018) · Zbl 1384.58020 |
| [27] | Logunov, A.; Malinnikova, E., Nodal sets of Laplace eigenfunctions: estimates of the Hausdorff measure in dimension two and three, (50 Years with Hardy Spaces. 50 Years with Hardy Spaces, Oper. Theory Adv. Appl. (2018)), 333-344 · Zbl 1414.31004 |
| [28] | Logunov, A.; Malinnikova, E.; Nadirashvili, N.; Nazarov, F., The Landis conjecture on exponential decay · Zbl 1486.35011 |
| [29] | Meshov, V., On the possible rate of decrease at infinity of the solutions of second-order partial differential equations, Math. USSR Sb., 72, 2, 343-361 (1992) · Zbl 0782.35010 |
| [30] | Rüland, A., On quantitative unique continuation properties of fractional Schrodinger equations: doubling, vanishing order and nodal domain estimates, Trans. Am. Math. Soc., 369, 2311-2362 (2017) · Zbl 1377.35276 |
| [31] | Vessella, S., Carleman estimates, optimal three cylinder inequality, and unique continuation properties for solutions to parabolic equations, Commun. Partial Differ. Equ., 28, 637-676 (2003) · Zbl 1024.35021 |
| [32] | Yau, S. T., Seminar on Differential Geometry, vol. 102 (1982), Princeton University Press · Zbl 0471.00020 |
| [33] | Zhu, J., Quantitative uniqueness for elliptic equations, Am. J. Math., 138, 733-762 (2016) · Zbl 1357.35103 |
| [34] | Zhu, J., Quantitative uniqueness of solutions to parabolic equations, J. Funct. Anal., 275, 9, 2373-2403 (2018) · Zbl 1401.35148 |
| [35] | Zhu, J., Doubling property and vanishing order of Steklov eigenfunctions, Commun. Partial Differ. Equ., 40, 8, 1498-1520 (2015) · Zbl 1323.47017 |
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