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contactarme por correo electrónico anes del 6 de enero
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Sobre el curso:
El objetivo principal del curso es introducir al estudiante a
la teoría lineal de ecuaciones diferenciales parciales basada
en espacios de Sobolev. Se discutirán: espacios de Hilbert y
de Banach, teoría de distribuciones, espacios de Sobolev,
ecuaciones elípticas, ecuaciones hiperbólicas y ecuaciones de
tipo parabólico.
Contenido:
- Temario del curso, calendario y bibliografía [PDF]
Material auxiliar:
Próximamente.Tareas:
- Tarea 1 [PDF].
Fecha de entrega: pasada.
- Tarea 2 [PDF].
Fecha de entrega: pasada.
- Tarea 3 [PDF].
Fecha de entrega: pasada.
Bibliografía
Bibliografía básica:
- L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, Amer. Math. Soc., Providence, RI, 1998.
- S. Salsa, Partial differential equations in action. From modelling to theory, Universitext, Springer-Verlag, Milan, Italia, 2008.
Bibliografía complementaria:
Análisis Funcional
- A. Bressan, Lecture Notes on Functional Analysis, vol. 143 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2013.
- S. Kesavan, Topics in functional analysis and applications, John Wiley & Sons, Inc., New York, 1989.
- K. Rektorys, Variational methods in mathematics,
science and engineering, D. Reidel Publishing Co.,
Dordrecht, second ed., 1980.
Teoría de distribuciones
- G. Eskin, Lectures on linear partial differential equations, vol. 123 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2011.
- L. Schwartz, Mathematics for the Physical Sciences, Addison-Wesley, 1966.
- R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, Studies in Advanced Mathematics, CRC-Press, Boca Ratón, Florida, 1994.
- A. H. Zemanian, Distribution theory and transform analysis: an introduction to generalized functions, with applications, Dover Publications, 1965.
Teoría de semigrupos
- K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
- K.-J. Engel and R. Nagel, A short course on operator
semigroups, Universitext, Springer-Verlag, New York,
2006.
- A. Pazy, Semigroups of linear operators and
applications to partial differential equations,
Springer verlag, 1983.
Espacios de Sobolev
- R. A. Adams, Sobolev spaces, Academic Press, 1975.
- H. Brezis, Functional Analysis, Sobolev spaces, and Partial Differential Equations, Universitext Springer Verlag, 2011.
- G. Eskin, Lectures on Linear Partial Differential
Equations, vol. 123 of Graduate Studies in
Mathematics, American Mathematical Society, Providence, RI,
2011.
- G. Leoni, A First Course in Sobolev spaces, vol.
181 of Graduate Studies in Mathematics, American
Mathematical Society, Providence, RI, second ed., 2017.
Ecuaciones elípticas
- R. Courant, D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations. Wiley Classics Library, John Wiley and Sons Inc., New York, 1962.
- G. B. Folland, Introduction to Partial Differential Equations, second ed., Princeton University Press, 1995.
- D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 1998.
- Q. Han, F. Lin, Elliptic partial differential equations, vol. 1 of Courant Lecture Notes in Mathematics, New York University Courant Institute of Mathematical Sciences, New York, 1997.
- R. E. Showalter, Hilbert space methods in partial differential equations, Electronic Journal of Differential Equations, Monograph no. 1, 1994.
Ecuaciones parabólicas
- O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uralceva, Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs vol. 23, American Mathematical Society, 1968.
- G. M. Lieberman, Second order parabolic partial differential equations, World Scientific, 1996.
- A. Friedman, Partial differential equations of parabolic type, Prentice Hall, 1964.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer verlag, 1983.
- M. Renardy, R. C. Rogers, An introduction to partial differential equations, second ed., vol. 13 of Texts in Applied Mathematics, Springer-Verlag, New York, 2004.
Ecuaciones hiperbólicas
- S. Alinhac, Hyperbolic partial differential equations, Universitext, Springer Verlag, 2009.
- S. Benzoni-Gavage, D. Serre, Multidimensional hyperbolic partial differential equations, first order systems and applications, Oxford University Press, 2007.
- O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, vol. 49 of Applied Mathematical Sciences, Springer Verlag, 1985.