Partial differential equations by John Fritz 4th Edition 1981

Identifies significant aspects of the theory and explores them with a limited
amount of machinery from mathematical analysis. This book contains a chapter
on Lewy's example of a linear equation without solutions.

Partial Differential Equations by Lawrence Evans 1998

This text gives a comprehensive survey of modern techniques in the theoretical
study of partial differential equations (PDEs) with particular emphasis on
nonlinear equations. The exposition is divided into three parts:
1) representation formulas for solutions,
2) theory for linear partial differential equations, and
3) theory for nonlinear partial differential equations.
Included are complete treatments of the method of characteristics; energy
methods within Sobolev spaces; regularity for second-order elliptic, parabolic,
and hyperbolic equations; maximum principles; the multidimensional calculus of
variations; viscosity solutions of Hamilton-Jacobi equations; shock waves and
entropy criteria for conservation laws; and much more.

The author summarizes the relevant mathematics required to understand
current research in PDEs, especially nonlinear PDEs. While he has reworked
and simplified much of the classical theory (particularly the method of
characteristics), he emphasizes the modern interplay between functional
analytic insights and calculus-type estimates within the context of Sobolev
spaces. Treatment of all topics is complete and self-contained. The book's
wide scope and clear exposition make it a suitable text for a graduate course
in PDEs.

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