Fichera's existence principle
In mathematics, and particularly in functional analysis, Fichera's existence principle is an existence and uniqueness theorem for solution of functional equations, proved by Gaetano Fichera in 1954.[1] More precisely, given a general vector space V and two linear maps from it onto two Banach spaces, the principle states necessary and sufficient conditions for a linear transformation between the two dual Banach spaces to be invertible for every vector in V.[2]
See also[]
- Banach fixed-point theorem
- Babuška–Lax–Milgram theorem
- Lax–Milgram theorem
- Lions–Lax–Milgram theorem
Notes[]
- ^ (Faedo 1957, p. 1), (Valent 1999, p. 84), (Leonardi, Passarelli di Napoli & Sbordone 2000, p. 221).
- ^ See (Fichera 1955, pp. 175–177, 1958, pp. 30–35), (Faedo 1957, pp. 1–2), (Miranda 1970, pp. 123–124), (Valent 1999, p. 84).
References[]
- ; (2000), "Some contributions of G. Fichera to the theory of Partial Differential Equations", in (ed.), Homage to Gaetano Fichera, , 7, , pp. 79–143, ISBN 978-88-7999-321-0, MR 1913527, Zbl 1005.35003. A survey of Gaetano Fichera's contributions to the theory of partial differential equations, written by two of his pupils.
- Faedo, Sandro (1957), "Su un principio di esistenza nell'analisi lineare" [On an existence principle in linear analysis], , Serie 3 (in Italian), 11 (1–2): 1–8, MR 0096957, Zbl 0087.32304.
- Faedo, Sandro (1958), "Applicazione ai problemi di derivata obliqua di un principio esistenziale e di una legge di dualità fra le formule di maggiorazione" [Application to oblique derivative problems of an existence principle and a duality law between estimates], Rendiconti di Matematica e delle sue Applicazioni, V Serie (in Italian), 16: 515–532, doi:10.1007/978-3-642-10918-8_4, ISBN 978-3-642-10916-4, MR 0096959, Zbl 0105.29902.
- Fichera, Gaetano (1962) [1954], Lezioni sulle trasformazioni lineari. Volume I: Introduzione all'analisi lineare [Lectures on linear transformations. Volume I: Introduction to linear analysis] (in Italian) (3rd reprint ed.), Roma: , pp. XIX+502, MR 0067346, Zbl 0057.33601: for a review of the book, see (1954), "G. Fichera, Lezioni sulle trasformazioni lineari, Vol. I: Introduzione all'Analisi lineare, Istituto Matematico dell'Università di Trieste, 1954 – pag. XVII + 502.", , Serie 3 (in Italian), 9 (4): 457–459.
- Fichera, Gaetano (1955), "Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni alle derivate parziali lineari", in Fichera, G. (ed.), Convegno Internazionale sulle Equazioni Lineari alle Derivate Parziali – Trieste 25–28 Agosto 1954 (in Italian), Roma: Edizioni Cremonese, pp. 174–227, MR 0074665, Zbl 0068.31101. The paper Some recent developments of the theory of boundary value problems for linear partial differential equations describes Fichera's approach to a general theory of boundary value problems for linear partial differential equations through a theorem similar in spirit to the Lax–Milgram theorem.
- Fichera, Gaetano (1958), Premesse ad una teoria generale dei problemi al contorno per le equazioni differenziali [Premises to a general theory of boundary value problems for differential equations], Corsi dell'Istituto Nazionale di Alta Matematica (in Italian), Roma: , pp. III+292. A monograph based on lecture notes, taken by and of a course held by Gaetano Fichera at the INdAM: for a review of the book, see Miranda, Carlo (1959), "G. Fichera, Premesse ad una teoria generale dei problemi al contorno per le equazioni differenziali, Libreria Eredi V., Roma", , Serie 3 (in Italian), 14 (4): 568–570.
- Leonardi, Salvatore; Passarelli di Napoli, Antonia; (2000), "On Fichera's existence principle in functional analysis", in (ed.), Problemi attuali dell'analisi e della fisica matematica. Atti del 2° simposio internazionale. Dedicato alla memoria del Prof. Gaetano Fichera. Taormina, 15–17 ottobre 1998 [Current problems in analysis and mathematical physics. Papers of the 2nd international symposium dedicated to the memory of Prof. Gaetano Fichera. Taormina, 15–17 ottobre 1998], Roma: , pp. 221–234, doi:10.4399/978887999264017 (inactive 31 October 2021), ISBN 978-88-7999-264-0, MR 1809690, Zbl 0956.00046CS1 maint: DOI inactive as of October 2021 (link).
- (1972), Theory of partial differential equations, Mathematics in Science and Engineering, 93, New York and London: Academic Press, pp. XIV+283, ISBN 0-12-449550-8, MR 0355280, Zbl 0245.35001, reviewed also by Schechter, Martin (October 1973), "Theory of Partial Differential Equations, by H. Melvin Lieberstein", SIAM Review, 15 (4): 809, doi:10.1137/1015120, JSTOR 2028753, and by Appleyard, David F. (March 1973), "Telegraphic Reviews. Theory of Partial Differential Equations. H. Melvin Lieberstein", The American Mathematical Monthly, 80 (3): 340, JSTOR 2318482
- Miranda, Carlo (1970) [1955], Partial Differential Equations of Elliptic Type, Ergebnisse der Mathematik und ihrer Grenzgebiete – 2 Folge, Band 2, translated by Motteler, Zane C. (2nd Revised ed.), Berlin – Heidelberg – New York: Springer Verlag, pp. XII+370, ISBN 978-3-540-04804-6, MR 0284700, Zbl 0198.14101.
- (1979), "Su un principio generale di esistenza in analisi lineare" [On a general existence principle in linear analysis], Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali, Series VIII (in Italian), 66 (5): 331–337, Zbl 0476.46002.
- (1999), "Existence problems", in Capriz, Gianfranco; ; (eds.), Interactions between Analysis and Mechanics. The Legacy of Gaetano Fichera. Convegno internazionale (Roma, 22--23 aprile 1998), Atti dei Convegni Lincei, 148, Roma: Accademia Nazionale dei Lincei, pp. 83–98, ISSN 0391-805X. An expository paper detailing the contributions of Gaetano Fichera and his school on the problem of numerical calculation of eigenvalues for general differential operators.
Categories:
- Partial differential equations
- Theorems in functional analysis