LECTURE NOTES IN ANALYSIS (2011) Sergiu Klainerman

LECTURE NOTES IN ANALYSIS ( 2011 ) Sergiu KlainermanDepartment of Mathematics, princeton University, princeton NJ 08544E-mail 1 INTRODUCTION TO PDE1. The world of PDETo start with partial differential equations, just like ordinary differential or integralequations, arefunctional equations. That means that the unknown, or unknowns,we are trying to determine are functions. In the case of partial differential equa-tions (PDE) these functions are to be determined from equations which involve, inaddition to the usual operations of addition and multiplication, partial derivativesof the functions. Below are the most basic examples, (Laplace equation) u= 0(1)where u= 2 x2u+ 2 y2u+ 2 z2u. The other two examples described in the sectionof fundamental mathematical definitions are (Heat Equation) tu+k u= 0,(2) (Wave equation) 2tu+c2 u= 0.(3)In both cases one is asked to find a functionu, depending on the variablest,x,y,z,which verifies the corresponding equations.

LECTURE NOTES IN ANALYSIS (2011) Sergiu Klainerman Department of Mathematics, Princeton University, Princeton NJ 08544 E-mail address: seri@math.princeton.edu. Part 1 INTRODUCTION TO PDE. 1. The world of PDE To start with partial di erential equations, just like ordinary di erential or integral

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