Lectures on elliptic and parabolic equations in Sobolev spaces.

*(English)*Zbl 1147.35001
Graduate Studies in Mathematics 96. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4684-1/hbk). xviii, 357 p. (2008).

The excellent lectures, written by one of the leading experts in the field, appear as the result of a two-quarter or an one-semester graduate course given by Nikolai V. Krylov at the Moscow State University and the School of Mathematics, University of Minnesota. These concentrate on some basic facts and ideas of the modern theory of linear elliptic and parabolic partial differential equations in Sobolev spaces. The author has perfectly reached his goal to show that this theory is based on some general and extremely powerful ideas and some simple computations. The main objects of the lectures are the Cauchy problem for parabolic equations and the first boundary-value problem for elliptic equations, with some guidelines regarding other boundary-value problems such as the Neumann or the oblique derivative problems or problems involving higher order elliptic operators acting on the boundary. The presentation is given in a way that after having followed the book the reader should acquire a good understanding of the wide variety of results and techniques.

The lectures start with the \(L_2\) theory of elliptic second-order equations in the whole space, first developing it for the Laplace operator on the basis of the Fourier transform. After that, the \(L_2\) theory of equations with variable coefficients is elaborated by using standard analytic tools such as partitions of unity, freezing the coefficients, a priori estimates and the method of continuity. With the aid of similar machinery, the author presents in Chapter 2 the \(L_2\) theory of second-order parabolic equations, concentrating itself mainly on equations in the whole space and the Cauchy problem. Chapter 3 has auxiliary character and is devoted to real analysis tools helping the passage from \(L_2\)- to \(L_p\) theory with \(p\neq2.\) In Chapter 4 the basic \(L_p\) estimates are derived first for parabolic and then for elliptic equations. Although the estimates in the elliptic case turn out to follow immediately from the parabolic ones, the author outlines how to do this directly on the level of a few exercises. Chapter 5 is devoted to the \(L_p\) theory of elliptic and parabolic equations with continuous coefficients in the whole space. Chapter 6 deals with the same issues for equations with \(VMO\) coefficients, while Chapter 7 treats parabolic equations with \(VMO\) coefficients in Sobolev spaces with mixed norms. As in Chapter 2, everything is done only for the equations in the whole space or for the Cauchy problem for equations whose coefficients are only measurable in the time variable.

Starting from Chapter 8, the considerations concentrate on elliptic equations in \(\mathbb R^d=\{x = (x_1,\ldots, x_d)\colon\;x_i\in(-\infty,\infty)\}\) or in domains \(\Omega\subset \mathbb R^d,\) even if almost everything proved for the elliptic equations in Chapter 8 is easily extendible to parabolic equations in \(\mathbb R\times\Omega.\) In particular, Chapter 8 deals with the \(W^2_p(\Omega)\) theory of second-order elliptic equations while Chapter 9 presents regularity results in \(W^k_p(\Omega).\) Chapter 10 collects results about Sobolev embedding theorems in \(W^k_p(\Omega),\) Morrey’s lemma, Gagliardo–Nirenberg and Kondrashov theorems. In Chapter 11 second-order elliptic equations of the form \(Lu-\lambda u=f\) are considered with particular attention paid on the properties of the resolvent operator and the decay estimates for solutions to \(Lu=f\) in \(\mathbb R^d.\) Chapter 12 is devoted to the Fourier transform and its relation to elliptic operators. Basic results are presented which regard pseudo-differential operators, elliptic boundary-value problems, existence of Green’s functions and estimates for them. The final Chapter 13 exposes the theory of elliptic and parabolic equations in the spaces \(H^\gamma_p\) of Bessel potentials.

These lecture notes are designed as a textbook and contain a great deal of carefully chosen exercises which help the reader to test himself. Prerequisites are basics of measure theory, the theory of \(L_p\) spaces and the Fourier transform.

Highly recommended text, addressed both to students and researchers working in the areas of partial differential equations, stochastic equations, operator theory, etc.

The lectures start with the \(L_2\) theory of elliptic second-order equations in the whole space, first developing it for the Laplace operator on the basis of the Fourier transform. After that, the \(L_2\) theory of equations with variable coefficients is elaborated by using standard analytic tools such as partitions of unity, freezing the coefficients, a priori estimates and the method of continuity. With the aid of similar machinery, the author presents in Chapter 2 the \(L_2\) theory of second-order parabolic equations, concentrating itself mainly on equations in the whole space and the Cauchy problem. Chapter 3 has auxiliary character and is devoted to real analysis tools helping the passage from \(L_2\)- to \(L_p\) theory with \(p\neq2.\) In Chapter 4 the basic \(L_p\) estimates are derived first for parabolic and then for elliptic equations. Although the estimates in the elliptic case turn out to follow immediately from the parabolic ones, the author outlines how to do this directly on the level of a few exercises. Chapter 5 is devoted to the \(L_p\) theory of elliptic and parabolic equations with continuous coefficients in the whole space. Chapter 6 deals with the same issues for equations with \(VMO\) coefficients, while Chapter 7 treats parabolic equations with \(VMO\) coefficients in Sobolev spaces with mixed norms. As in Chapter 2, everything is done only for the equations in the whole space or for the Cauchy problem for equations whose coefficients are only measurable in the time variable.

Starting from Chapter 8, the considerations concentrate on elliptic equations in \(\mathbb R^d=\{x = (x_1,\ldots, x_d)\colon\;x_i\in(-\infty,\infty)\}\) or in domains \(\Omega\subset \mathbb R^d,\) even if almost everything proved for the elliptic equations in Chapter 8 is easily extendible to parabolic equations in \(\mathbb R\times\Omega.\) In particular, Chapter 8 deals with the \(W^2_p(\Omega)\) theory of second-order elliptic equations while Chapter 9 presents regularity results in \(W^k_p(\Omega).\) Chapter 10 collects results about Sobolev embedding theorems in \(W^k_p(\Omega),\) Morrey’s lemma, Gagliardo–Nirenberg and Kondrashov theorems. In Chapter 11 second-order elliptic equations of the form \(Lu-\lambda u=f\) are considered with particular attention paid on the properties of the resolvent operator and the decay estimates for solutions to \(Lu=f\) in \(\mathbb R^d.\) Chapter 12 is devoted to the Fourier transform and its relation to elliptic operators. Basic results are presented which regard pseudo-differential operators, elliptic boundary-value problems, existence of Green’s functions and estimates for them. The final Chapter 13 exposes the theory of elliptic and parabolic equations in the spaces \(H^\gamma_p\) of Bessel potentials.

These lecture notes are designed as a textbook and contain a great deal of carefully chosen exercises which help the reader to test himself. Prerequisites are basics of measure theory, the theory of \(L_p\) spaces and the Fourier transform.

Highly recommended text, addressed both to students and researchers working in the areas of partial differential equations, stochastic equations, operator theory, etc.

Reviewer: Dian K. Palagachev (Bari)

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35Jxx | Elliptic equations and elliptic systems |

35Kxx | Parabolic equations and parabolic systems |

35R05 | PDEs with low regular coefficients and/or low regular data |