Summary
We consider the classical problem of finding the density ϱ of a material body Ω embedded into a region S, when the potential generated by Ω (possibly coinciding with S) is known outside (or on the surface of) S. In the set of such solutions we look for the density\(\bar \varrho\)which has the smallestL 2-norm and we prove that\(\bar \varrho\) belongs toL 2= H (Ω), the space of square summable functions harmonic in Ω. However\(\bar \varrho\)is unstable, i.e. itsL 2-norm does not depend continuously upon the L2-norm of the potential. We show how a continuous dependence may be restored by introducing mild restrictions on the set of admissible solutions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.Anger,Convex sets in inverse problems, Trudy Vsesojuznoj Konferencii po Uravnenijam s Castnymi Proizvodnymi, Moscow (1978) (Russian).
G.Anger,Lectures on potential theory and inverse problems, Summer School in Geophysics, Bergakademie Freiberg (1978).
M. M. Lavrent'ev,Some improperly posed problems of Mathematical Physics, Izdat. Sibirsk. Otdel. Akad. Nauk. SSSR, Novosibirsk (1962); English translation: Springer Tracts in Natural Philosophy,11, Springer (1967).
L. A. Ljusternik -V. I. Sobolev,Elements of functional analysis, Hindustan Publ. Delhi (1961).
J.Nečas,Les méthodes directes en théorie des équations elliptiques, Masson (1967).
L. Nirenberg -H. Walker,The null spaces of elliptic differential equations in R n, J. Math. Anal. Appl.,42 (1973), pp. 271–301.
A. I. Prilepko,The uniqueness of a solution of an inverse problem represented by an integral equation of first kind, Dokl. Akad. Nauk. SSSR,167 (1966), pp. 751–754; English translation: Soviet Math. Dokl.,7 (1966), pp. 471–475.
A. I. Prilepko,On the uniqueness of the determination of the form of a body by the values of the exterior potential, Dokl. Akad. Nauk. SSSR,160 (1965), pp. 40–43.
B. W.Schulze - G.Wildenhain,Methoden der Potentialtheorie für elliptische Differentialgleichungen beliebiger Ordung, Birkhäuser (1977).
F.Sansò,Internal Collocation, Mem. Accad. Naz. Lincei,16 (1980), fasc. 1 (1981).
K.Yosida,Functional Analysis, Springer (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lorenzi, A., Pagani, C.D. An inverse problem in potential theory. Annali di Matematica pura ed applicata 129, 281–303 (1981). https://doi.org/10.1007/BF01762147
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01762147