An inverse problem for the equation u=-cu-d
Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1181-1209.

Let Ω be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem

u=-cu-dinΩ,u=0onΩ.

We prove two results: 1) If Ω is not a ball and if one considers only solutions with -cu-d0, then there exist at most finitely many pairs of coefficients (c,d) so that the normal derivative u ν| Ω equals a given ψ0.

2) If one imposes no sign condition on the solutions but one additionally supposes that Ω is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients (c,d) so that u ν| Ω equals a given non-degenerate ψ. Our analysis is related to work on the Pompeiu–Schiffer conjectures. To illustrate this relation we also show how our analysis provides a very elementary and short proof of a result, due to Berenstein, concerning the Schiffer conjecture.

On considère un domaine plan convexe Ω, dont la courbure du bord n’est pas trop dégénérée. Dans cet article, nous donnons des réponses partielles à une question d’identification liée au problème aux limites

u=-cu-dinΩ,u=0onΩ.

Nous montrons deux résultats : 1) Si Ω n’est pas une boule et si on considère seulement des solutions telles que -cu-d0, alors il existe au plus un nombre dénombrable de paires de coefficients (c,d), telles que la dérivée normale u ν| Ω soit égale à une fonction ψ0 donnée.

2) Si aucune condition de signe n’est imposée, mais si on fait l’hypothèse supplémentaire que Ω est suffisamment différent d’une boule, alors, de nouveau, il existe au plus un nombre dénombrable de paires de coefficients (c,d), telles que u ν| Ω soit égale à une fonction non-dégénérée ψ donnée. Notre analyse est reliée à des travaux sur les conjectures de Pompeiu–Schiffer. Pour illustrer cette relation, nous montrons aussi comment notre analyse permet de montrer de manière très simple et rapide un résultat dû à Berenstein, concernant la conjecture de Schiffer.

@article{AIF_1994__44_4_1181_0,
     author = {Vogelius, Michael},
     title = {An inverse problem for the equation $\triangle u=-cu-d$},
     journal = {Annales de l'Institut Fourier},
     pages = {1181--1209},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {44},
     number = {4},
     year = {1994},
     doi = {10.5802/aif.1429},
     zbl = {0813.35136},
     mrnumber = {95h:35246},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1429/}
}
TY  - JOUR
AU  - Vogelius, Michael
TI  - An inverse problem for the equation $\triangle u=-cu-d$
JO  - Annales de l'Institut Fourier
PY  - 1994
DA  - 1994///
SP  - 1181
EP  - 1209
VL  - 44
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.1429/
UR  - https://zbmath.org/?q=an%3A0813.35136
UR  - https://www.ams.org/mathscinet-getitem?mr=95h:35246
UR  - https://doi.org/10.5802/aif.1429
DO  - 10.5802/aif.1429
LA  - en
ID  - AIF_1994__44_4_1181_0
ER  - 
%0 Journal Article
%A Vogelius, Michael
%T An inverse problem for the equation $\triangle u=-cu-d$
%J Annales de l'Institut Fourier
%D 1994
%P 1181-1209
%V 44
%N 4
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.1429
%R 10.5802/aif.1429
%G en
%F AIF_1994__44_4_1181_0
Vogelius, Michael. An inverse problem for the equation $\triangle u=-cu-d$. Annales de l'Institut Fourier, Volume 44 (1994) no. 4, pp. 1181-1209. doi : 10.5802/aif.1429. http://www.numdam.org/articles/10.5802/aif.1429/

[1] C.A. Berenstein, An inverse spectral theorem and its relation to the Pompeiu problem, Journal d'Analyse Mathématique, 37 (1980), 128-144. | MR | Zbl

[2] C.A. Berenstein and P.C. Yang, An inverse Neumann problem, J. reine angew. Math., 382 (1987), 1-21. | MR | Zbl

[3] E. Beretta and M. Vogelius, An inverse problem originating from magnetohydrodynamics, Arch. Rational Mech. Anal., 115 (1991), 137-152. | MR | Zbl

[4] E. Beretta and M. Vogelius, An inverse problem originating from magnetohydrodynamics III. Domains with corners of arbitrary angles, Submitted to Asymptotic Analysis. | Zbl

[5] M. Berger and B. Gostiaux, Differential Geometry : Manifolds, Curves and Surfaces, Springer-Verlag, New York, 1988. | MR | Zbl

[6] J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, Wiley/Gauthier-Villars, New York, 1989. | MR | Zbl

[7] L. Brown and J.-P. Kahane, A note on the Pompeiu problem for convex domains, Math. Ann., 259 (1982), 107-110. | MR | Zbl

[8] L. Brown, B.M. Schreiber and B.A. Taylor, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier, 23-3 (1973), 125-154. | Numdam | MR | Zbl

[9] N. Garofalo and F. Segala, Another step toward the solution of the Pompeiu problem in the plane, Commun. in Partial Differential Equations, 18 (1993), 491-503. | MR | Zbl

[10] B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. | MR | Zbl

[11] F. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974. | MR | Zbl

[12] S.I. Pohožaev, Eigenfunctions of the equation Δu + λf(u) = 0, Soviet Math. Dokl., 6 (1965), 1408-1411 (English translation of Dokl. Akad. Nauk. SSSR, 165 (1965), 33-36). | Zbl

[13] M.H. Protter and H.F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. | MR | Zbl

[14] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. | MR | Zbl

[15] S.A. Williams, A partial Solution of the Pompeiu problem, Math. Ann., 223 (1976), 183-190. | MR | Zbl

[16] S.A. Williams, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana University Mathematics Journal, 30 (1981), 357-369. | MR | Zbl

Cited by Sources: