On the Lax-Milgram Theorem

Témavezető:	Kovács Sándor
	ELTE, Numerikus Analı́zis Tanszék
email:	alex@ludens.elte.hu

Projekt leírás

It is well known that a generalization of the Riesz-Fréchet theorem (Riesz's representation theorem) is the Lax-Milgram theorem, which is a fundamental result in functional ana-lysis that provides a necessary and sufficient condition for the existence and uniqueness of solutions to certain types of partial differential equations. During the semester, the candidate is expected to give an overview of possible generalizations of the theorem and their applications, based on the literature below and on the literature agreed with the supervisors. The work carried out during the project may form part of a MSc thesis or a research outcome for Scientific Student Associations' Conference.

Előfeltételek

Familiarity with the basics of functional analysis and measure theory, reading research articles in English.

Hivatkozások

[1] De Carli, Laura; Vellucci, Pierluigi: \textit{Applications of Lax-Milgram theorem to problems in frame theory}, Sampl. Theory Signal Process. Data Anal. (2023) {\bf 21}(1) 18 pp.

[2] Edward M. Landesman: \textit{A Generalized Lax-Milgram Theorem}, Proceedings of the American Mathematical Society {\bf 19}(2) (1968) {\bf 11}(1) 339--344.

[3] Fechner, Włodzimierz: \textit{Functional inequalities motivated by the Lax-Milgram theorem}, J. Math. Anal. Appl. (2013) {\bf 40}(2) 411--414.

[4] Kov\'acs, S.: \href{http://numanal.inf.elte.hu/~alex/hu/anyag/PROGINF/FunkAnal/FunkAnalKS.pdf}{\textit{Funkcionálanalízis feladatokban}}, egyetemi jegyzet, Budapest, 2013.

[5] Lipcsey, Zs.: \textit{A Lax-Milgram-tétel általánosítűása}, MTA SZTAKI KÖZLEMÉNYEK , {\bf 8} (1972), 93--101. ISSN 0133-7459

[6] Lin, Bor Luh; Lohman, Robert H.: \textit{The Lax-Milgram theorem for topological vector spaces}, J. Math. Anal. Appl., {\bf 40} (1972), 601--608.

[7] Nyamoradi, Nemat; Hamidi, Mohammad Rassol.: \textit{An extension of the Lax-Milgram theorem and its application to fractional differential equations}, Electron. J. Differential Equations {\bf 95} (2015), 1--9.

[8] Roşca, Ioan: \textit{Bilinear coercive and weakly coercive operators}, An. Univ. Bucureşti Mat. (2002) {\bf 51}(2) 183--188.

[9] Roşca, Ioan: \textit{On the solvability of linear variational equations with constrains}, An. Univ. Bucureşti Mat. (2003) {\bf 52}(1) 65--74.

[10] Zeidler, E.: \textit{Applied Functional Analysis, Applications to Mathematical Physics}, Springer Verlag, 1995.