Introduction of Elliptic PDES (Han-Lin)

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Outline
1
C O U R A N T
1
Q I N G H A N
F A N G H U A L I N
LECTURE
NOTES
Elliptic Partial
Dierential
Equations
American Mathematical Society
Courant Institute of Mathematical Sciences
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Contents
Contents 3
Preface 5
1 Harmonic Functions 7
1.1 Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Mean Value Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Epilogue 15
2.1 Non-Dierentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Smooth Manifolds with Additional Structure . . . . . . . . . . . . . . . 18
2.3 Generalized Cohomology Theories . . . . . . . . . . . . . . . . . . . . . 19
Appendices 23
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4 CONTENTS
preface
In the fall of 1992, the second author gave a course called “Intermediate P.D.E.” at
the Courant Institute. The purpose of that course was to present some basic methods
for obtaining various a priori estimates for second-order partial dierential equations of
elliptic type with particular emphasis on maximal principles Harnack inequalities, and
their applications. The equations one deals with are always linear, although they also
obviously apply to nonlinear problems. Students with some knowledge of real variables
and Sobolev functions should be able to follow the course without much diculty.
In 1992, the lecture notes were taken by the rst author. In 1995 at the University
of Notre Dame, the rst author gave a similar course. The original notes were then
much extended, resulting in their present form It is not our intention to give a complete
account of the related theory. Our goal is simply to provide these notes as a bridge
between the elementary book of F. John [9],which also studies equations of other types,
and the somewhat advanced book of D. Gilbarg and N. Trudinger [8] which gives a
relatively complete account of the theory of elliptic equations of second order. We also
hope our notes can serve as a bridge between the recent elementary book of N. Krylov
[11] on the classical theory of elliptic equations developed before or around the 1960s
and the book by Caarelli and Cabré [4] which studies fully nonlinear elliptic equations,
the theory obtained in the 1980s.
The authors wish to thank Karen Jacobs, Cheryl Hu, Joan Hoerstman, and Daisy
Calderon for the wonderful typing job. The work was also partially supported by Na-
tional Science Foundation Grants DMS No. 9401546 and DMS No. 9501122.
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