Partial Differential Equations
This book is based on a course I have given five times at the University of Michigan, beginning in 1973. The aim is to present an introduction to a sampling of ideas, phenomena, and methods from the subject of partial differential equations that can be presented in one semester and requires no previous knowledge of differential equations. The problems, with hints and discussion, form an important and integral part of the course. In our department, students with a variety of specialties-notably differen tial geometry, numerical analysis, mathematical physics, complex analysis, physics, and partial differential equations-have a need for such a course. The goal of a one-term course forces the omission of many topics. Everyone, including me, can find fault with the selections that I have made. One of the things that makes partial differential equations difficult to learn is that it uses a wide variety of tools. In a short course, there is no time for the leisurely development of background material. Consequently, I suppose that the reader is trained in advanced calculus, real analysis, the rudiments of complex analysis, and the language offunctional analysis. Such a background is not unusual for the students mentioned above. Students missing one of the "essentials" can usually catch up simultaneously. A more difficult problem is what to do about the Theory of Distributions.
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Algebraic approximation asserts belongs Cauchy data Cauchy problem Cauchy sequence characteristic Choose compact compute converges coordinates Corollary curves defined Definition denoted dense Dirichlet boundary conditions Dirichlet problem Discussion domain eigenfunction elliptic equal equivalent estimate example exponentially extends uniquely Figure finite follows formula Fourier transform given heat equation Hilbert space Hint Hl(il holomorphic hypersurface identity implies inequality infinity initial value problem integral integrand L2 norm Lemma linear map method neighborhood noncharacteristic nonlinear norm Note partial differential equations partial differential operator polynomial proof of Theorem propagator Proposition Prove real analytic Regularity Theorem result right-hand side satisfies Schrodinger equation sequence smooth solution solvability solve subset suffices to show supp Suppose tangential tends to zero term Theory topology uniformly bounded vanishes variable vector field wave equation yields