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An inverse problem for the equation $$\Delta u=-cu-d$$. (English) Zbl 0813.35136
Summary: Let $$\Omega$$ 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 $\Delta u = -cu-d \;\text{in}\;\Omega,\quad u=0 \;\text{on}\;\partial \Omega.$ We prove two results:
1) If $$\Omega$$ is not a ball and if one considers only solutions with $$-cu-d\leq 0$$, then there exist at most finitely many pairs of coefficients $$(c,d)$$ so that the normal derivative $$\partial u/ \partial \nu|_{ \partial \Omega}$$ equals a given $$\psi \neq 0$$.
2) If one imposes no sign condition on the solutions but one additionally supposes that $$\Omega$$ is sufficiently far from being a ball, then there exist again at most finitely many pairs of coefficients $$(c,d)$$ so that $$\partial u/ \partial \nu|_{ \partial \Omega}$$ equals a given non- degenerate $$\psi$$.
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.

##### MSC:
 35R30 Inverse problems for PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
##### Keywords:
Radon transform; stationary phase; Schiffer conjecture
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##### References:
  C.A. BERENSTEIN, An inverse spectral theorem and its relation to the Pompeiu problem, Journal d’Analyse Mathématique, 37 (1980), 128-144. · Zbl 0449.35024  C.A. BERENSTEIN and P.C. YANG, An inverse Neumann problem, J. reine angew. Math., 382 (1987), 1-21. · Zbl 0623.35078  E. BERETTA and M. VOGELIUS, An inverse problem originating from magnetohydrodynamics, Arch. Rational Mech. Anal., 115 (1991), 137-152. · Zbl 0732.76096  E. BERETTA and M. VOGELIUS, An inverse problem originating from magnetohydrodynamics III. domains with corners of arbitrary angles, Submitted to Asymptotic Analysis. · Zbl 0853.76093  M. BERGER and B. GOSTIAUX, Differential geometry : manifolds, curves and surfaces, Springer-Verlag, New York, 1988. · Zbl 0629.53001  J. BLUM, Numerical simulation and optimal control in plasma physics, Wiley/Gauthier-Villars, New York, 1989. · Zbl 0717.76009  L. BROWN and J.-P. KAHANE, A note on the Pompeiu problem for convex domains, Math. Ann., 259 (1982), 107-110. · Zbl 0464.30035  L. BROWN, B.M. SCHREIBER and B.A. TAYLOR, Spectral synthesis and the Pompeiu problem, Ann. Inst. Fourier, 23-3 (1973), 125-154. · Zbl 0265.46044  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. · Zbl 0818.35136  B. GIDAS, W.-M. NI and L. NIRENBERG, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. · Zbl 0425.35020  F. OLVER, Asymptotics and special functions, Academic Press, New York, 1974. · Zbl 0303.41035  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 0141.30202  M.H. PROTTER and H.F. WEINBERGER, Maximum principles in differential equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1967. · Zbl 0153.13602  J. SERRIN, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. · Zbl 0222.31007  S.A. WILLIAMS, A partial solution of the Pompeiu problem, Math. Ann., 223 (1976), 183-190. · Zbl 0329.35045  S.A. WILLIAMS, Analyticity of the boundary for Lipschitz domains without the Pompeiu property, Indiana University Mathematics Journal, 30 (1981), 357-369. · Zbl 0439.35046
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