There are long books and short books. It is hard to say which kind is more valuable， or which kind one should read. When a short book contains all essential things of a subject and arranges them in a clear and accessible way， a short book is probably more preferable for some obvious reasons. Additionally， ifit is written by aleading expert on the subjects and a master expositor， then the answer is a definite and clear yes.
The booklet "Existence Theorems in Partial Differential Equations" is of this type. It was written by the world top expert on partial differential equations， Louis Nirenberg， at one of the peaks of his long and productive life. It covers existence and uniqueness of solutions of elliptic differential equations. When one opens this booklet or rather lecture notes， one can immediately see the flow ofthoughts ofa great mathematician：it is direct to the point， everything moves smoothly and quickly， and there is no unnecessary discussions or digressions. Elliptic differential equations are central in partial differential equations and their applications in differential geometry. Though many results have been obtained in the past half century， the essential things are still the same. Furthermore， though there have been many books on differential equations， the freshness and the spirit of these lecture notes cannot be surpassed by later more comprehensive ones.
目錄：
Part I Estence Theorems in Partial Differential Equations
1 Prelinunaries
1．1 Introduction
1．2 The Mamum Principle
1．3 Consequences of the Mamum Principle
2 The Potential Equation
2．1 Fundamental Solution
2．2 The Poisson Integral Formula
2．3 The Mean Value Property of Potential Functions
2．4 Estimates of Derivatives of Harmonic Functions and Analyticity
2．5 The Theorems and Inequality of Harnack
2．6 Theorem on Removable Singularities
3 The Perron Method for Solving the Dirichlet Problem
3．1 The Perron Method
3．2 The Perron Method for More General Elliptic Equations
4 Schauder Estimates
4．1 Poisson''s Equation
4．2 A Preliminary Estimate
4．3 Statement of Schauder''s Estimates
4．4 Some Applications of the Interior Estimates
4．5 The Boundary Value Problem
4．6 Strong Barrier Functions， and the Boundary Value Problem
5 Derivation of the Schauder Estimates
5．1 A Preliminary Estimate
5．2 A Furtherlnvestigation of the Poisson Equation
5．3 Completion of the Interior Estimates
……
Part II Seminar on Differential Geometry in the Large
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