Limiting absorption principles for the Navier equation in elasticity

Barceló, Juan Antonio; Folch-Gabayet, Magali; Pérez-Esteva, Salvador; Ruiz, Alberto; Vilela, Mari Cruz

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 11 (2012) no. 4, pp. 817-842.

We prove some a priori estimates for the resolvent of Navier equation in elasticity, when one approaches the spectrum (Limiting Absorption Principles). They are extensions of analogous estimates for the resolvent of the euclidean Laplacian in ${ℝ}^n$. As a consequence, we get some results for the evolution equation and for the spectral measure.

Classification : 35J47, 74B05, 42B37

[1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa (4) 2 (1975), 151–218. | EuDML | Numdam | MR | Zbl

[2] S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math. 30 (1976), 1–38. | MR | Zbl

[3] J. A. Barceló and A. Córdoba, Band-limited functions: $L^p$-convergence, Trans. Amer. Math. Soc. (2) 313 (1989), 655-669. | MR | Zbl

[4] J. A. Barceló, A. Ruiz and L. Vega, Weighted estimates for the Helmholtz equation and some applications, J. Funct. Anal. 150 (1997), 356–382. | MR | Zbl

[5] A. P. Calderón and A. Zygmund, On singular integrals, Amer. J. Math. 78 (1956), 289–309. | MR | Zbl

[6] S. Campanato, Proprietà di una famiglia di spazi funzionali, Ann. Scuola Norm. Sup. Pisa 18 (1964), 137–160. | EuDML | Numdam | MR | Zbl

[7] F. Chiarenza and M. Frasca, A remark on a paper by C. Fefferman, Proc. Amer. Math. Soc. 108 (1990), 407–409. | MR | Zbl

[8] F. Chiarenza and A. Ruiz, Uniform $L^2-$weighted Sobolev inequalities, Proc. Amer. Math. Soc. 112 (1991), 53–64. | MR | Zbl

[9] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. | EuDML | MR | Zbl

[10] D. Colton and R. Kress, “Inverse Acoustic and Electromanetic Scattering Theory”, Springer-Verlag, New York, 1992. | MR | Zbl

[11] J. Duoandikoetxea Zuazo, “Fourier Analysis”, AMS, Graduate Studies in Mathematics, V 29, 2001. | MR | Zbl

[12] J. García-Cuerva and J. L. Rubio de Francia, “Weighted Norms Inequalities and Related Topics”, North-Holland, Amsterdam, 1985. | MR | Zbl

[13] C. Herz, Lipschitz spaces and Bernstein’s theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968), 283–324. | MR | Zbl

[14] G. Hu, S. Lu and D. Yang, Boundedness of rough singular integral operators on homogeneous Herz spaces, J. Austr. Math. Soc. 66 (1999), 201–223. | MR | Zbl

[15] T. Kato, Wave operators and similarity for non-selfadjoint operators, Math. Ann. 162 (1966), 258–279. | EuDML | MR | Zbl

[16] T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phy. (4) 1 (1989), 481–496. | MR | Zbl

[17] C. Kenig, G. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 255–288. | EuDML | Numdam | MR | Zbl

[18] C. Kenig, A. Ruiz and C. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), 329–347. | MR | Zbl

[19] V. D. Kupradze, “Three-Dimensional Problems of Elasticity and Thermoelasticity”, North-Holland, Amsterdam, 1979.

[20] R. Leis, “Initial Boundary Value Problems in Mathematical Physics”, John Wiley & Sons, New York, 1986. | MR | Zbl

[21] X. Li and D. Yang, Boundedness of some sublinear operators on Herz spaces, Illinois J. Math. 40 (1996), 484–501. | MR | Zbl

[22] B. Perthame and L. Vega, Morrey-Campanato estimates for Helmholtz equations, J. Funct. Anal. 164 (1999), 340–355. | MR | Zbl

[23] A. Ruiz, Recovery of the singluarities of a potential from fixed angle scattering data, Comm. Partial Differential Equations 26 (2001), 1721–1738. | MR | Zbl

[24] A. Ruiz, Harmonic analysis and inverse problems, http://web.uam.es/gruposinv/inversos/publicaciones/Inverseproblems.pdf

[25] A. Ruiz and A. Vargas, Partial recovery of a potential from backscattering data, Comm. Partial Differential Equations 30 (2005), 67–96. | MR | Zbl

[26] A. Ruiz and L. Vega, Unique continuation for Schrödinger equations with potential in Morrey classes, Publ. Math. 35 (1991), 291–298. | EuDML | MR | Zbl

[27] A. Ruiz and L. Vega, On local regularity of Schrödinger equations, Duke Math. J. 76 (1994), 913–940. | MR | Zbl

[28] A. Ruiz and L. Vega, Local regularity of solutions to wave equations with time-dependent potentials, Int. Math. Res. Not. INRN 1 (1993), 13–27. | MR | Zbl

[29] G. Stampacchia, ${ℒ}^{p,\lambda }$-Spaces and interpolation, Comm. Pure Appl. Math. 17 (1964), 293–306. | MR | Zbl

[30] L.Vega and N. Visciglia, On the local smoothing for the Schrödinger equation, Proc. Amer. Math. Soc. 135 (2007), 119–128. | MR | Zbl