This book is based on notes from a beginning graduate course on partial differential equations. Prerequisites for using the book are a solid undergraduate course in real analysis. There are more than 100 exercises in the book. Some of them are just exercises, whereas others, even though they may require new ideas to solve them, provide additional important information about the subject.

It is a great pleasure to see this book—written by a great master of the subject—finally in print. This treatment of a core part of mathematics and its applications offers the student both a solid foundation in basic calculations techniques in the subject, as well as a basic introduction to the more general machinery, e.g., distributions, Sobolev spaces, etc., which are such a key part of any modern treatment. As such this book is ideal for more advanced undergraduates as well as mathematically inclined students from engineering or the natural sciences. Shubin has a lovely intuitive writing style which provides a gentle introduction to this beautiful subject. Many good exercises (and solutions) are provided!

—Rafe Mazzeo, Stanford University

This text provides an excellent semester's introduction to classical and modern topics in linear PDE, suitable for students with a background in advanced calculus and Lebesgue integration. The author intersperses treatments of the Laplace, heat, and wave equations with developments of various functional analytic tools, particularly distribution theory and spectral theory, introducing key concepts while deftly avoiding heavy technicalities.

—Michael Taylor, University of North Carolina, Chapel Hill

Readership

Graduate students interested in partial differential equations.

• Mark Anthony Armstrong, Basic topology, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1979 original. MR 705632
• V. I. Arnol′d, Ordinary differential equations, MIT Press, Cambridge, Mass.-London, 1978. Translated from the Russian and edited by Richard A. Silverman. MR 0508209
• V. I. Arnol′d, Mathematical methods of classical mechanics, Graduate Texts in Mathematics, vol. 60, Springer-Verlag, New York-Heidelberg, 1978. Translated from the Russian by K. Vogtmann and A. Weinstein. MR 0690288
• F. A. Berezin and M. A. Shubin, Schrödinger equation, Kluwer, Dordrecht, 1987.
• R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York, Vol. I, 1953; Vol. II, 1962.
• Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
• Gregory Eskin, Lectures on linear partial differential equations, Graduate Studies in Mathematics, vol. 123, American Mathematical Society, Providence, RI, 2011. MR 2809923, DOI 10.1090/gsm/123
• R. Feynman, R. B. Leighton, and N. Sands, The Feynman lectures on physics, Springer-Verlag, Reading, MA, Palo Alto, London, 1967.
• David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
• Alfred Gray, Tubes, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. MR 1044996
• Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
• Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
• Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Band 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0161012
• Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
• L. Hörmander, The analysis of partial differential operators. v. I, II, Springer, Berlin et al., 1983.
• F. John, Partial differential equations, Springer, New York et al., 1986.
• A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Dover Publications, Inc., Mineola, New York, 1999.
• N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR 1406091, DOI 10.1090/gsm/012
• Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
• N. N. Lebedev, Special functions and their applications, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR 0350075
• Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
• J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
• V. Palamodov, Topics in mathematical physics, Spring semester, preprint, 2002.
• I. G. Petrovskii, Partial differential equations, Iliffe Books Ltd., London; distributed by W. B. Saunders Co., Philadelphia, Pa., 1967. Translated from the Russian by Scripta Technica, Ltd. MR 0211021
• Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
• A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
• Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
• Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
• Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469
• Laurent Schwartz, Mathematics for the physical sciences, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0207494
• Laurent Schwartz, Analyse, Collection Enseignement des Sciences, No. 11, Hermann, Paris, 1970 (French). Deuxième partie: Topologie générale et analyse fonctionnelle. MR 0467223
• Georgi E. Shilov, Generalized functions and partial differential equations, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1968. Authorized English edition revised by the author; Translated and edited by Bernard D. Seckler. MR 0230129
• Norio Shimakura, Partial differential operators of elliptic type, Translations of Mathematical Monographs, vol. 99, American Mathematical Society, Providence, RI, 1992. Translated and revised from the 1978 Japanese original by the author. MR 1168472, DOI 10.1090/mmono/099
• S. L. Sobolev and E. R. Dawson, Partial differential equations of mathematical physics, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. Translated from the third Russian edition by E. R. Dawson; English translation edited by T. A. A. Broadbent. MR 0178220
• M. E. Taylor, Partial differential equations, Vol I, Basic theory; Vol. II, Qualitative studies of linear equations, Springer-Verlag, New York, 1996.
• F. Treves, Topological vector spaces, distributions and kernels, Academic Press, Inc., (London, New York), LTD, third printing, 1970.
• V. S. Vladimirov, Equations of mathematical physics, “Mir”, Moscow, 1984. Translated from the Russian by Eugene Yankovsky [E. Yankovskiĭ]. MR 764399
• Frank W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971 edition. MR 722297
• John Wermer, Potential theory, 2nd ed., Lecture Notes in Mathematics, vol. 408, Springer, Berlin, 1981. MR 634962

• Mark Anthony Armstrong, Basic topology, corrected reprint of the 1979 original, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1983. MR 705632
• V. I. Arnol′d, Ordinary differential equations, translated from the Russian and edited by Richard A. Silverman, MIT Press, Cambridge, Mass.-London, 1978. MR 0508209
• V. I. Arnol′d, Mathematical methods of classical mechanics, translated from the Russian by K. Vogtmann and A. Weinstein; Graduate Texts in Mathematics, 60, Springer-Verlag, New York-Heidelberg, 1978. MR 0690288
• F. A. Berezin and M. A. Shubin, Schrödinger equation, Kluwer, Dordrecht, 1987.
• R. Courant and D. Hilbert, Methods of mathematical physics, Interscience, New York, Vol. I, 1953; Vol. II, 1962.
• Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
• Gregory Eskin, Lectures on linear partial differential equations, Graduate Studies in Mathematics, vol. 123, American Mathematical Society, Providence, RI, 2011. MR 2809923, DOI 10.1090/gsm/123
• R. Feynman, R. B. Leighton, and N. Sands, The Feynman lectures on physics, Springer-Verlag, Reading, MA, Palo Alto, London, 1967.
• David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, reprint of the 1998 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2001. MR 1814364
• Alfred Gray, Tubes, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1990. MR 1044996
• Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
• Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
• Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0161012
• Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
• L. Hörmander, The analysis of partial differential operators. v. I, II, Springer, Berlin et al., 1983.
• F. John, Partial differential equations, Springer, New York et al., 1986.
• A. N. Kolmogorov and S. V. Fomin, Elements of the theory of functions and functional analysis, Dover Publications, Inc., Mineola, New York, 1999.
• N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR 1406091, DOI 10.1090/gsm/012
• Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, translated from the Russian by Scripta Technica, Inc., translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
• N. N. Lebedev, Special functions and their applications, revised edition, translated from the Russian and edited by Richard A. Silverman; unabridged and corrected republication, Dover Publications, Inc., New York, 1972. MR 0350075
• Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR 1817225, DOI 10.1090/gsm/014
• J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, translated from the French by P. Kenneth; Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. MR 0350177
• V. Palamodov, Topics in mathematical physics, Spring semester, preprint, 2002.
• I. G. Petrovskii, Partial differential equations, translated from the Russian by Scripta Technica, Ltd, Iliffe Books Ltd., London (distributed by W. B. Saunders Co., Philadelphia, Pa.), 1967. MR 0211021
• Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
• A. P. Robertson and W. J. Robertson, Topological vector spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 53, Cambridge University Press, New York, 1964. MR 0162118
• Walter Rudin, Principles of mathematical analysis, 3rd ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976. MR 0385023
• Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
• Helmut H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1966. MR 0193469
• Laurent Schwartz, Mathematics for the physical sciences, Hermann, Paris; Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0207494
• Laurent Schwartz. Analyse (French), Deuxième partie: Topologie générale et analyse fonctionnelle; Collection Enseignement des Sciences, No. 11, Hermann, Paris, 1970. MR 0467223
• Georgi E. Shilov, Generalized functions and partial differential equations, authorized English edition revised by the author, translated and edited by Bernard D. Seckler, Gordon and Breach, Science Publishers, Inc., New York-London-Paris, 1968. MR 0230129
• Norio Shimakura, Partial differential operators of elliptic type, translated and revised from the 1978 Japanese original by the author, Translations of Mathematical Monographs, vol. 99, American Mathematical Society, Providence, RI, 1992. MR 1168472
• S. L. Sobolev and E. R. Dawson, Partial differential equations of mathematical physics, translated from the third Russian edition by E. R. Dawson; English translation edited by T. A. A. Broadbent, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. MR 0178220
• M. E. Taylor, Partial differential equations, Vol I, Basic theory; Vol. II, Qualitative studies of linear equations, Springer-Verlag, New York, 1996.
• F. Treves, Topological vector spaces, distributions and kernels, Academic Press, Inc., (London, New York), LTD, third printing, 1970.
• V. S. Vladimirov, Equations of mathematical physics, translated from the Russian by Eugene Yankovsky [E. Yankovskiĭ], “Mir”, Moscow, 1984. MR 764399
• Frank W. Warner, Foundations of differentiable manifolds and Lie groups, corrected reprint of the 1971 edition, Graduate Texts in Mathematics, vol. 94, Springer-Verlag, New York-Berlin, 1983. MR 722297
• John Wermer, Potential theory, 2nd ed., Lecture Notes in Mathematics, vol. 408, Springer, Berlin, 1981. MR 634962