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admits a unique apply assume Banach space boundary condition bounded domain bounded linear C(fi Cauchy data Cauchy problem Chapter characteristic curves classical solution coefficients conclude constant converges convex critical point defined denote depends derivatives differential equations Dirichlet problem distribution eigenfunctions eigenvalues elliptic operator example Exercises for Section fact Figure finite finite-dimensional first-order fixed point Fourier Fredholm function theorem fundamental solution harmonic heat equation Hilbert space hyperbolic implies inequality initial condition initial value problem Lemma Let us consider linear functional linear operator maximum principle method Moreover Neumann nonlinear norm Notice obtain PDEs proof of Theorem prove quasilinear rarefaction regularity replace Riesz representation theorem satisfies second-order Section 2.3 semigroup semilinear sequence smooth Sobolev spaces solve subharmonic sufficiently small Suppose unique solution variables vector verify wave equation weak solution
Page iii - ... course in partial differential equations which I have taught at Northeastern University many times since 1980. The first term is intended to give the student a basic and classical introduction to the subject, including first-order equations by the method of characteristics and linear second @ order equations which arise in mathematical physics: the wave equation, Laplace equation, and heat equation. All of this material is more than adequately covered by many textbooks which are readily available....
Page iii - ... many textbooks which are readily available. The second term, however, is intended to introduce the student to a wide variety of more modern methods, especially the use of functional analysis, which has characterized much of the recent development of partial differential equations. This latter material is not as readily available, except in a number of specialized reference books. This textbook is intended to bridge this gap by providing the student with a basic introduction to the subject and...