Abstract
We use Pitt inequalities for the Fourier transform to prove the following weighted gradient inequality
This inequality is a Carleman-type estimate that yields unique continuation results for solutions of first order differential equations and systems.
Similar content being viewed by others
References
Barostichi, R.F., Cordaro, P.D., Petronilho, G.: Strong unique continuation for systems of complex vector fields. Bull. Sci. Math. 138(4), 457–469 (2014)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)
Caffarelli, L., Kohn, R., Nirenberg, L.: First order interpolation inequalities with weight. Compos. Math. 53, 259–275 (1984)
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)
Ciarlet, P.G., Mardare, C.: On rigid and infinitesimal rigid displacements in shell theory. J. Math. Pures Appl. (9) 83(1), 1–15 (2004)
Cnops, J.: An Introduction to Dirac Operators on Manifolds. Springer, Berlin (2012)
Cosner, C.: On the definition of ellipticity for systems of partial differential equations. J. Math. Anal. Appl. 158(1), 80–93 (1991)
Cuesta, M., Ramos Quoirin, H.: A weighted eigenvalue problem for the \(p\)-Laplacian plus a potential. Nonlinear Differ. Equ. Appl. 16, 469–491 (2009)
Douglis, A.: Uniqueness in Cauchy problems for elliptic systems of equations. Commun. Pure Appl. Math. 6, 291–298 (1953)
Dupaigne, L.: Stable Solutions of Elliptic Partial Differential Equations, Monographs and Surveys in Pure and Applied Mathematics. CRC Press, Boca Raton (2011)
De Carli, L., Edward, J., Hudson, S., Leckband, M.: Minimal support results for Schrödinger’s equation. Forum Math. 27(1), 343–371 (2015)
De Carli, L., Gorbachev, D., Tikhonov, S.: Pitt inequalities and restriction theorems for the Fourier transform. Rev. Mat. Iberoam. 33(3), 789–808 (2017)
De Carli, L., Hudson, S.: Geometric remarks on the level curves of harmonic functions. Bull. Lond. Math. Soc. 42(1), 83–95 (2010)
De Carli, L., Nacinovich, M.: Unique continuation in abstract pseudoconcave CR manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(1), 27–46 (1998)
Douglis, A., Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Commun. Pure Appl. Math. 8, 503–538 (1955)
De Carli, L., Okaji, T.: Strong unique continuation for the Dirac operator. Publ. Res. Inst. Math. Sci. 35(6), 825–846 (1999)
Edward, J., Hudson, S., Leckband, M.: Existence problems for the \(p\)-Laplacian. Forum Math. 27(2), 1203–1225 (2013)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Fabes, E., Kenig, C., Serapioni, R.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equ. 7(1), 77–116 (1982)
Fall, M.M.: Semilinear Elliptic Equations for the Fractional Laplacian with Hardy Potential. arXiv:1109.5530v4 (2011)
Fall, M.M., Felli, V.: Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete Contin. Dyn. Syst. 35(12), 5827–5867 (2015)
Ferrari, F., Valdinoci, E.: Some weighted Poincaré inequalities. Indiana Univ. Math. J. 58(4), 1619–1637 (2009)
Filippucci, R., Pucci, P., Rigoli, M.: Nonlinear weighted \(p\)-Laplacian elliptic inequalities with gradient terms. Commun. Contemp. Math. 12(3), 501–535 (2010)
Hayashida, K.: Unique continuation theorem of elliptic systems of partial differential equations. Proc. Jpn. Acad. 38, 630–635 (1962)
Heinig, H.P.: Weighted Sobolev inequalities for gradients. In: Harmonic Analysis and Applications. Applied and Numerical Harmonic Analysis, pp. 17–23. Birkhäuser, Springer, Cham, Switzerland (2006)
Heinig, H.P.: Weighted norm inequalities for classes of operators. Indiana Univ. Math. J. 33(4), 573–582 (1984)
Herbst, I.W.: Spectral theory of the operator \( (p ^2 + m^2)^{ 1/2} -Ze^2/r\). Commun. Math. Phys. 53(3), 285–294 (1977)
Horiuchi, T.: Best constant in weighted sobolev inequality with weights being powers of distance from the origin. J. Inequal. Appl. 1, 275–292 (1997)
Hile, G.N., Protter, M.H.: Unique continuation and the Cauchy problem for first order systems of partial differential equations. Commun. Partial Differ. Equ. 1(5), 437–465 (1976)
Jerison, D.: Carleman inequalities for the Dirac and Laplace operators and unique continuation. Adv. Math. 62(2), 118–134 (1986)
Jerison, D., Kenig, C.: Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. 121, 463–494 (1985)
Jurkat, W., Sampson, G.: On rearrangement and weight inequalities for the Fourier transform. Indiana Univ. Math. J. 33, 257–270 (1984)
Kalf, H., Yamada, O.: Note on the paper: “Strong unique continuation property for the Dirac equation” by L. De Carli and T. Ōkaji. Publ. Res. Inst. Math. Sci. 35(6), 847–852 (1999)
Khafagy, S.A.: On positive weak solution for a nonlinear system involving weighted \(p\)-Laplacian. J. Adv. Res. Dyn. Control Syst. 4(4), 50–58 (2012)
Kenig, C., Ruiz, A., Sogge, C.D.: Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. 55(2), 329–347 (1987)
Koch, H., Tataru, D.: Carleman estimates and absence of embedded eigenvalues. Commun. Math. Phys. 267(2), 419–449 (2006)
Koch, H., Tataru, D.: Recent results on unique continuation for second order elliptic equations. In: Carleman Estimates and Applications to Uniqueness and Control Theory (Cortona, 1999), Progress in Nonlinear Differential Equations and Their Applications, vol. 46. pp. 73–84. Birkhäuser Boston, Boston, MA (2001)
Lakey, J.D.: Weighted Fourier transform inequalities via mixed norm Hausdorff–Young inequalities. Can. J. Math. 46(3), 586–601 (1994)
Lankeit, J., Neff, P., Pauly, D.: Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity. C. R. Math. Acad. Sci. Paris 351(5–6), 247–250 (2013)
Levitan, B.M., Sargsjan, I.S.: Introduction to Spectral Theory: Self Adjoint Ordinary Differential Operators, Translations of Mathematical Monographs, vol. 39. American Mathematical Society, Providence (1975)
Long, R., Nie, F.: Weighted Sobolev inequality and eigenvalue estimates of Schrödinger operators. Lect. Notes Math. 1494, 131–141 (1990)
Maz’ya, V.G.: Sobolev Spaces. Springer, Berlin (1985)
Muckenhoupt, B.: Weighted norm inequalities for the Fourier transform. Trans. Am. Math. Soc. 276, 729–742 (1983)
Okaji, T.: Strong unique continuation property for elliptic systems of normal type in two independent variables. Tohoku Math. J. (2) 54(2), 309–318 (2002)
Okaji, T.: Strong unique continuation property for first order elliptic systems. In: Carleman Estimates and Applications to Uniqueness and Control Theory (Cortona, 1999), Progress in Nonlinear Differential Equations and Their Applications, vol. 46, pp. 149–164. Birkhäuser Boston, Boston, MA (2001)
Pérez, C.: Sharp \(L^p\)-weighted Sobolev inequalities. Ann. Inst. Fourier (Grenoble) 45(3), 809–824 (1995)
Ruf, B.: Superlinear elliptic equations and systems. In: Chipot, M. (ed.) Handbook of Differential Equations. Stationary Partial Differential Equations, vol. 5, pp. 211–276. Elsevier, Amsterdam (2008)
Sawyer, E.T.: A characterization of two weight norm inequalities for fractional fractional and Poisson integrals. Trans. Am. Math. Soc. 308, 533–545 (1988)
Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)
Simon, B.: On positive eigenvalues of one-body Schrödinger operators. Commun. Pure Appl. Math. 22, 531–538 (1969)
Simon, B.: Schrodinger semigroups. Bull. Am. Math. Soc. (N.S.) 7(3), 447–526 (1982)
Sogge, C.D.: Strong uniqueness theorems for second order elliptic differential equations. Am. J. Math. 112(6), 943–984 (1990)
Sinnamon, G.: A weighted gradient inequality. Proc. R. Soc. Edinb. Sect. A 111(3–4), 329–335 (1989)
Tamura, M.: A note on strong unique continuation for normal elliptic systems with Gevrey coefficients. J. Math. Kyoto Univ. 49(3), 593–601 (2009)
Tataru, D.: Unique continuation problems for partial differential equations. In: Geometric Methods in Inverse Problems and PDE Control, The IMA Volumes in Mathematics and Its Applications, vol. 137. pp. 239–255. Springer, New York (2004)
Uryu, H.: The local uniqueness of some characteristic Cauchy problems for the first order systems. Funkcialaj Ekvacioj 38, 21–36 (1995)
Wolff, T.: Recent work on sharp estimates in second order elliptic unique continuation problems. In: Garcia-Cuerva, J., Hernandez, E., Soria, F., Torrea, J.L. (eds.) Fourier Analysis and Partial Differential Equations. Studies in Advanced Mathematics, pp. 99–128. CRC Press, Boca Raton (1995)
Yang, Q., Lian, B.: On the best constant of weighted Poincaré inequalities. J. Math. Anal. Appl. 377(1), 207–215 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
D. G. was supported by the Russian Science Foundation under Grant 18-11-00199. S. T. was partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programme of the Generalitat de Catalunya.
Rights and permissions
About this article
Cite this article
De Carli, L., Gorbachev, D. & Tikhonov, S. Weighted gradient inequalities and unique continuation problems. Calc. Var. 59, 89 (2020). https://doi.org/10.1007/s00526-020-1716-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-020-1716-8