Some Aspects in the PDE. The Dirichlet and the Cauchy Problems
DOI:
https://doi.org/10.17268/sel.mat.2024.01.10Keywords:
Potential, Newton, Cuachy, Nirenberg, Hörmander, Problem and principle od Dirichlet, Cauchy problemAbstract
In this article we present two classic problems in PDE: the Dirichlet problem and the Cauchy problem. Some ideas are previously discussed, such as functions and identities’s Green, and “bad positions” problems. The work of L.Nirenberg [1] is discussed in detail where some results of L.Hörmander [2] are used.
References
Nirenberg L. Uniqueness in Cauchy problems for differential equations with constant leading coefficients. Communications on Pure and Applied Mathematics. 1957;X.
Hörmander L. On the theory of general partial differential operators. 1955;94.
Lewy H. An example of a smooth linear partial differential equation without solution. Annals of Mathematics. 1957;66(1):155-8.
Plis A. Non-uniqueness in Cauchy’s problem for differential equations of elliptic type. Journal of Mathematics and Mechanics. 1960;9(4):557-62.
Plis A. A smooth linear elliptic differential equation without any solution in a sphere. Communications on Pure and Applied Mathematics. 1961;14(3):599-617.
Figueiredo D. Teoría clássica do potencial. Brasilia: Universidade de Brasília; 1963.
Greenspan D. Introduction to partial differential equations. New York: McGraw-Hill; 1961.
Epstei B. Partial differential equations, An introduction. New York: McGraw-Hill; 1962.
Hellwig G. Partial differential equations. An introduction. New York: Blaisdell Pub.Com; 1964.
Myint-U T. Partial differential equations of mathematical physics. New York: American Elsevier Publi. Comp.; 1973.
Courant R, Hilbert D. Methods of mathematical physics. vol. II. New York: Interscience Publishers; 1962.
Figueiredo D. Analise de Fourier e equac¸oes diferenciais parciais. Rio de Janeiro: IMPA. CNPq; 1977.
Aleksandrov AD, et al ANK. La matemática: su contenido, métodos y significado. vol. 2. Madrid: Alíanza Editorial; 1974.
Kline M. El Pensamiento matemático de la Antigüedad a nuestros días. Madrid: Alianza Editorial; 2012.
Ortiz A. Aspectos básicos en ecuaciones en derivadas parciales. Notas de Matemática No 3 Universidad Nacional de Trujillo. 1988.
Ortiz A. Tópicos sobre ecuaciones en derivadas parciales. Dpto Matemática Universidad Nacional de Trujillo. 2004.
Figueiredo D. O Princípio de Dirichlet. Matemática Universitaria SBM. 1985;1.
Coddington EA. An introduction to ordinary differential equations. New York: Prentice- Hall, INC; 1961.
Evans LC. Partial differential equations. Graduate Studies in Mathematics. vol. 19. American Mathematical Society; 2010.
Hörmander L. Linear partial differential operators. 1963.
Dieudonné J. History of functional analysis. North-Holland Math Studies. 1983;77.
Brézis H. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer Science & Business Media; 2010.
Bers L, John F, Schechter M. Partial differential equations. New York: Interscience Publishers. John Wiley Sons, Inc; 1964.
Stakgold I. Green’s functions and boundary value problems. New York: John Wiley -Sons; 1979.
Egorov YV, Shubin MA. Partial differential equations I. Foundations of the classical theory. New York: Springer-Verlag; 1992.
Jost J. Partial differential equations. Graduate Texts in Mathematics. Springer - Verlag; 2002.
Jr CM. Some recent developments in the theory of partial differential equations. Bulletin of AMS. 1962;68(4).
Brezis H, Browder F. Partial differential equations in the 20th Century. 1998;Advances in Mathematics 135.
Hörmander L. Linear differential operators. Actes, Congres Intern Math. 1970;1.
Treves F. On local solvability of linear partial differential equations. Indiana: Purdue University, Lafayette; 1969.
Calderón AP, Zygmund A. On the existence of certain singular integrals. Acta Mathematica. 1952;88.
Maurin K. Methods of Hilbert spaces. Polish Scientific Publishers; 1967.
Pedersorn RN. Uniqueness in Cauchy’s problem for elliptic equations with double characteristics. Arkiv för Matematik. 1966;6(27):535-49.
Chung SY. Uniqueness of the Cauchy problem for certain quasi-linear partial differential equations. Journal of the Korea Society of Mathematical Education. 1986;25(1):33-7.
Zeman M. Uniqueness of solutions of the Cauchy problem for linear partial differential equation with characteristics of variable multiplicity. Journal of Differential Equations. 1978;27(1):1-18.
Calderón AP. Uniqueness in the Cauchy problem for partial differential equations. American Journal Mathematics. 1958;80(1):16-36.
Nirenberg L. On elliptic partial differential equations. vol. 13. Annali della Scuola Normale Superiore di Pisa; 1959.
Vásquez JM. El Problema de Dirichlet. Universidad de Panamá; 1989.
Ponce AC. II Escola de E.D. Rio de Janeiro. In: Métodos Clássicos em Teoría do Potencial; 2006.
Schwartz L. Méthodes mathématiques pour les sciences physiques. Hermann S G Paris.
;VI(115).
Mijailov VP. Ecuaciones diferenciales en derivadas parciales. Moscú: Editorial MIR; 1978.
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