\(L^p\)-\(L^q\) estimates for electromagnetic Helmholtz equation. (English) Zbl 1234.35047

Garcia, Andoni

J. Math. Anal. Appl. 384, No. 2, 409-420 (2011).

Summary: In space dimension \(n\geq 3\), we consider the electromagnetic Schrödinger Hamiltonian \(H=(\nabla -iA(x))^2-V\) and the corresponding Helmholtz equation \[ (\nabla-iA(x))^2u+u-V(x)u=f\quad \text{in }\mathbb{R}^n. \] We extend the well-known \(L^p\)-\(L^q\) estimates for the solution of the free Helmholtz equation to the case when the electromagnetic Hamiltonian \(H\) is considered.

MSC:

35B45	A priori estimates in context of PDEs
35Q60	PDEs in connection with optics and electromagnetic theory
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

dispersive equations; Helmholtz equation; magnetic potential

References:

[1]	Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. Sc. Norm. Super. Pisa, 2, 2, 151-218 (1975) · Zbl 0315.47007
[2]	Agmon, S.; Hörmander, L., Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30, 1-38 (1976) · Zbl 0335.35013
[3]	Burq, N.; Planchon, F.; Stalker, J. G.; Tahvildar-Zadeh, A. S., Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., 203, 519-549 (2003) · Zbl 1030.35024
[4]	Carbery, A.; Soria, F., Almost-everywhere convergence of Fourier integrals for functions in Sobolev spaces, and an \(L^2\)-localisation principle, Rev. Mat. Iberoamericana, 4, 2, 319-337 (1988) · Zbl 0692.42001
[5]	DʼAncona, P.; Fanelli, L.; Vega, L.; Visciglia, N., Endpoint Strichartz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258, 3227-3240 (2010) · Zbl 1188.81061
[6]	Fanelli, L., Non-trapping magnetic fields and Morrey-Campanato estimates for Schrödinger operators, J. Math. Anal. Appl., 357, 1-14 (2009) · Zbl 1170.35374
[7]	S. Gutiérrez, Un problema de contorno para la ecuación de Ginzburg-Landau, PhD thesis, Basque Country University, 2000.; S. Gutiérrez, Un problema de contorno para la ecuación de Ginzburg-Landau, PhD thesis, Basque Country University, 2000.
[8]	Gutiérrez, S., Non trivial \(L^q\) solutions to the Ginzburg-Landau equation, Math. Ann., 328, 1-25 (2004) · Zbl 1109.35045
[9]	Ikebe, T.; Saito, Y., Limiting absorption method and absolute continuity for the Schrödinger operator, J. Math. Kyoto Univ., 12, 513-542 (1972) · Zbl 0257.35022
[10]	Kenig, C. E.; Ruiz, A.; Sogge, D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55, 329-347 (1987) · Zbl 0644.35012
[11]	Kenig, C. E.; Ponce, G.; Vega, L., Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré, 10, 255-288 (1993) · Zbl 0786.35121
[12]	Regbaoui, R., Strong uniqueness for second order differential operators, J. Differential Equations, 141, 201-217 (1997) · Zbl 0887.35040
[13]	Ruiz, A.; Vega, L., On local regularity of Schrödinger equations, Int. Math. Res. Not., 13-27 (1993) · Zbl 0812.35016