×

On the strong unique continuation principle for inequalities of Maxwell type. (English) Zbl 0703.35028

For inequalities of Maxwell type \[ | curl a| +| div(\alpha a)| \leq cr^{\epsilon -1}| a| \] there is proved the strong unique continuation principle; i.e. a vanishes identically in a neighborhood of zero if a decays of infinite order at zero. The definite matrix valued function \(\alpha\) (x) satisfies the hypothesis \[ | \partial_ j\alpha (x)| \leq c| x|^{\epsilon -1}\quad (j=1,2,3) \] for some \(\epsilon >0\). In the case of a real function \(\alpha\) (x) the weaker condition \(| \partial_ r\alpha (x)| \leq c| x|^{\epsilon -1}\) is sufficient.
The proof relies on Carleman inequalities in \(L^ 2\) and suitable representations of curl and divergence in polar coordinates including especially the perturbation \(\alpha\).
Reviewer: V.Vogelsang

MSC:

35B60 Continuation and prolongation of solutions to PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
35R45 Partial differential inequalities and systems of partial differential inequalities
78A25 Electromagnetic theory (general)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pures Appl., IX. Ser.36, 235-249 (1957) · Zbl 0084.30402
[2] Alinhac, S., Baouendi, M.S.: Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities. Am. J. Math.102, 179-217 (1980) · Zbl 0425.35098 · doi:10.2307/2374175
[3] Aronszajn, N., Krzywicki, A., Szarski, J.: A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat.4, 417-453 (1962) · Zbl 0107.07803 · doi:10.1007/BF02591624
[4] Carleman, T.: Sur un probleme d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes. Ark. Mat. (17)26B, 1-9 (1939) · Zbl 0022.34201
[5] Cordes, H.O.: Über die Bestimmtheit der Lösungen elliptischer Differentialgleichungen durch Anfangsvorgaben. Nachr. Akad. Wiss. Gött. II. Math.-Phys. Kl. (11), 239-258 (1956) · Zbl 0074.08002
[6] Dautray, R., Lions, J.L.: Mathematical analysis and numerical methods for science and technology, vol. 3. Berlin Heidelberg New York: Springer 1990 · Zbl 0784.73001
[7] Heinz, E.: Über die Eindeutigkeit beim Cauchyschen Anfangswertproblem einer elliptischen Differentialgleichung zweiter Ordnung. Nachr. Akad. Wiss. Gött. II. Math.-Phys. Kl. (1), 1-12 (1955) · Zbl 0067.07503
[8] Hörmander, L.: Uniqueness theorems for second order elliptic differential equations. Commun. Partial Differ. Equations8, 21-64 (1983) · Zbl 0546.35023 · doi:10.1080/03605308308820262
[9] Jerison, D.: Carleman inequalities for the Dirac and Laplace operators and unique continuation. Adv. Math.62, 118-134 (1986) · Zbl 0627.35008 · doi:10.1016/0001-8708(86)90096-4
[10] Jerison, D., Kenig, C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. II. Ser.121, 463-494 (1985) · Zbl 0593.35119 · doi:10.2307/1971205
[11] Leis, R.: Über die eindeutige Fortsetzbarkeit der Lösungen der Maxwellschen Gleichungen in anisotropen, inhomogenen Medien. Bull. Polytechnic Inst. Jassy14(8), 119-124 (1968) · Zbl 0183.37801
[12] Leis, R.: Zur Theorie der elektromagnetischen Schwingungen in anisotropen, inhomogenen Medien. Math. Z.106, 213-224 (1968) · doi:10.1007/BF01110135
[13] Picard, R.: On the low frequency asymptotics in electromagnetic theory. J. Reine Angew. Math.354, 50-73 (1984) · Zbl 0541.35049 · doi:10.1515/crll.1984.354.50
[14] Vogelsang, V.: Über das lokale Verhalten der Lösungen elliptischer Differentialgleichungen zweiter Ordnung. Nachr. Akad. Wiss. Gött. II. Math.-Phys. Kl. (3), 19-27 (1982) · Zbl 0533.35020
[15] Vogelsang, V.: Absence of embedded eigenvalues of the Dirac equation for long range potentials. Analysis7, 259-274 (1987) · Zbl 0643.35073
[16] Vogelsang, V.: Absolutely continuous spectrum of Dirac operators for long range potentials. J. Funct. Anal.76, 67-86 (1988) · Zbl 0652.47030 · doi:10.1016/0022-1236(88)90049-3
[17] Vogelsang, V.: On the point spectrum of Maxwell operators in long range inhomogenous media (preprint 1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.