This book studies second order elliptic boundary value problems on bounded smooth domains in ${ℝ}^n$ from a functional analytic point of view. More precisely, the authors rely on Sobolev spaces, the theory of closed unbounded operators in Hilbert spaces, and entropy numbers. The traditional approach through elliptic forms is avoided in this book.

The book begins with the classical Dirichlet problem for the Laplace operator. The first chapter discusses harmonic functions, the Newton potential, Green's functions and the Poisson kernel for the ball.

The next three chapters collect some functional analytic prerequisites. Chapter 2 deals with distributions and, in particular, tempered distributions and the Fourier transform. Chapter 3 studies the Sobolev spaces $W_p^k$ on ${ℝ}^n$ and the half-space ${ℝ}_{+}^n$ along with the Fourier analytically defined scale $H^s\left({ℝ}^n\right)$ extending the $W_2^k$-spaces. Here density, embedding and extension theorems are proved, and the trace operator from $W_p^1\left({ℝ}_{+}^n\right)$ to $L_p\left(\partial {ℝ}_{+}^n\right)$ is introduced. The next chapter refines these considerations by looking at Sobolev spaces on bounded domains with a $C^{\infty }$-boundary. Again, density, embedding and extension theorems and the trace operator from $W_2^s\left(\text{Ω}\right)$ to $W_2^{s-1/2}\left(\partial \text{Ω}\right)$ are dicussed.

The heart of the book is Chapter 5. The authors investigate elliptic differential operators

on a bounded $C^{\infty }$-domain $\text{Ω}\subset {ℝ}^n$ with sufficiently smooth coefficients. The authors prove a priori estimates and Gårding's inequality, and they go on to show existence and uniqueness in the homogeneous and inhomogeneous Dirichlet problem for the operator $A+\lambda$ when $\lambda$ is large enough. For this, the authors use functional analytic techniques such as the Friedrichs extension of semi-bounded operators and the Riesz-Schauder theory of compact operators. They also treat the Neumann problem, but mainly for the Laplace operator. Also, the smoothness of the solution, measured in terms of the Sobolev scale, is investigated.

The general functional analytic methodology provides assertions about the eigenvalues of $A$ such as ${\lambda }_k\to \infty$ if $A$ is self-adjoint. Finer points of the eigenvalue distribution are considered in Chapter 7 that contains estimates like $|{\lambda }_k|\ge c{k}^{2/n}$, for instance. This is proved by means of Carl's inequality that relates the eigenvalues of a compact operator with its entropy numbers. The abstract theory of entropy numbers and approximation numbers is presented in Chapter 6, and entropy numbers of Sobolev embeddings are estimated. The spectral properties of elliptic operators then follow by means of factorisation techniques.

All the proofs in this book are very detailed, well-structured and accessible. There are several exercises throughout the text some of which are solved in the appendix. Each chapter starts with an overview of the basic ideas and finishes with a section of notes hinting at possible extensions of the results in the setting of Bessel potential spaces, Besov spaces and Triebel-Lizorkin spaces.

The present text is very suitable for readers who wish to study elliptic equations with the help of the elaborate theory of function spaces that has been developed over the past 30 years.