About This Course

This book is a first serious course in partial differential equations for a reader working alone. It is aimed at an advanced undergraduate or beginning graduate student in mathematics, physics, or engineering who wants to know where the classical equations come from, how their solutions behave, how to prove that behaviour, and how the modern notion of a weak solution grows out of the classical theory.

Four prerequisites are used constantly. From Multivariable Calculus: partial derivatives, the chain rule, multiple integrals, and the divergence theorem. From Differential Equations: solving linear ordinary differential equations with constant coefficients and the existence and uniqueness theorem for initial value problems. From Real Analysis: limits, continuity, uniform convergence, and compactness. From Linear Algebra: symmetric matrices, eigenvalues, and inner products. The last three chapters also use the Lebesgue integral and the spaces \(L^p\) from Measure Theory and one abstract theorem from Functional Analysis. A reader without measure theory can still follow those chapters by reading “integrable” as “integrable in the sense you know” and accepting the completeness of \(L^2\) ; every imported result is stated with the hypotheses the argument needs and is marked once with its library identifier. It is not proved again here.

Six abilities mark the reader who has finished the book, and none of them should need notes. Derive transport, diffusion, wave, and equilibrium equations from a conservation law and a constitutive assumption. Solve first-order equations by characteristics and say when and where the classical solution breaks. Prove the mean value property, the maximum principles, and uniqueness by energy, and use them to bound solutions you cannot compute. Write and use the representation formulas of d’Alembert, Poisson, and the heat kernel, and build series solutions by separation of variables. Decide whether a rough function solves an equation in the distributional or weak sense. Put an elliptic boundary value problem in weak form, verify the hypotheses of the Lax–Milgram theorem, and compute a Galerkin approximation by hand.

Notation is conventional. An open set \(\Omega\subseteq\mathbb R^n\) is the spatial domain, \(\partial\Omega\) its boundary, \(\overline\Omega\) its closure, and \(\nu\) the outward unit normal. Subscripts denote partial derivatives, so \(u_t=\partial u/\partial t\) and \(u_{xy}=\partial^2u/\partial y\,\partial x\) . The gradient \(\nabla u\) and the Laplacian \(\Delta u=u_{x_1x_1}+\dots+u_{x_nx_n}\) act on the space variables only. The ball of radius \(r\) about \(x\) is \(B_r(x)\) ; its volume is \(\lvert B_r\rvert\) and the surface area of its boundary is \(\lvert\partial B_r\rvert\) . We write \(\omega_n\) for the surface area of the unit sphere in \(\mathbb R^n\) , so \(\omega_2=2\pi\) , \(\omega_3=4\pi\) , \(\lvert\partial B_r\rvert=\omega_nr^{n-1}\) , and \(\lvert B_r\rvert=\omega_nr^n/n\) . The class \(C^k(\Omega)\) consists of functions with continuous partial derivatives up to order \(k\) in \(\Omega\) , and \(C^k(\overline\Omega)\) of those whose derivatives up to order \(k\) extend continuously to \(\overline\Omega\) .

Each chapter ends with exercises ranging from recall to proof. Work them with pencil first. The companion solutions give full arguments for some, graduated hints for others, and bare answers for a few. Several exercises present a plausible argument that is wrong and ask you to find the flaw; in this subject a formula applied outside its hypotheses usually still produces an answer, so that skill matters. The cumulative review mixes chapters without saying which tool each problem needs.