Elliptic and parabolic equations with discontinuous coefficients.

*(English)*Zbl 0958.35002
Mathematical Research. 109. Weinheim: Wiley-VCH. 256 p. DM 198.00; EUR 101.24 (2000).

This interesting book concerns strong solutions to second-order elliptic and parabolic equations and systems with discontinuous coefficients and with nondivergence principal part (unfortunately, it does not contain examples of such nondivergence form equations, arising from applications). The authors mainly consider homogeneous Dirichlet or oblique derivative problems on bounded convex domains \(\Omega\subset\mathbb{R}^n\) with \(C^2\)-smooth boundary and with zero initial conditions (in the parabolic case).

Chapter 1 deals with linear equations such that the principal coefficients \(a_{ij}\in L^\infty(\Omega)\) of the elliptic operator satisfy a Cordes condition (i.e. the eigenvalues of the matrix \((a_{ij})\) should not scatter too much) or the Sobolev regularity condition \(a_{ij}\in W^{1,n}(\Omega)\). Strong solvability in \(W^{2,p} (\Omega)\) (with \(p<2\) close to two) is shown as well as higher regularity of \(D_tu\) and \(D^2u\) in Morrey spaces and of \(u\) and all its derivatives in Sobolev spaces. The exceptional case \(n=2\), where no Cordes condition and no Sobolev regularity of the principal coefficient is needed, is presented in detail.

Chapter 2 concerns linear parabolic and semilinear elliptic equations with VMO principal coefficients. Here strong solvability is proved in \(W^{2,p}(\Omega)\) for all \(p>1\), using an explicit representation formula for \(D^2u\) in terms of singular integrals.

Chapter 3 is about quasilinear equations and systems which satisfy a nonlinear Cordes condition. This condition enables to apply Companatos theory of near operators (the operator are near to so-called basic nonvariational operators which play the same role in the quasilinear case as the Laplace and the heat operator play in the linear case). The authors show higher integrability of \(D_tu\) and \(D^2u\) and HĂ¶lder continuity of \(u\) or of \(Du\) or of \(D_tu\) and \(D^2u\) (depending on the space dimension \(n)\).

The book collects many techniques that are important in regularity theory (Miranda-Talenti estimates, Cacciopoli type estimates, Calderon-Zygmund kernels and their commutators). Hence, it will prove an essential reference for all those with an interest in solution regularity to elliptic and parabolic pdes’.

Chapter 1 deals with linear equations such that the principal coefficients \(a_{ij}\in L^\infty(\Omega)\) of the elliptic operator satisfy a Cordes condition (i.e. the eigenvalues of the matrix \((a_{ij})\) should not scatter too much) or the Sobolev regularity condition \(a_{ij}\in W^{1,n}(\Omega)\). Strong solvability in \(W^{2,p} (\Omega)\) (with \(p<2\) close to two) is shown as well as higher regularity of \(D_tu\) and \(D^2u\) in Morrey spaces and of \(u\) and all its derivatives in Sobolev spaces. The exceptional case \(n=2\), where no Cordes condition and no Sobolev regularity of the principal coefficient is needed, is presented in detail.

Chapter 2 concerns linear parabolic and semilinear elliptic equations with VMO principal coefficients. Here strong solvability is proved in \(W^{2,p}(\Omega)\) for all \(p>1\), using an explicit representation formula for \(D^2u\) in terms of singular integrals.

Chapter 3 is about quasilinear equations and systems which satisfy a nonlinear Cordes condition. This condition enables to apply Companatos theory of near operators (the operator are near to so-called basic nonvariational operators which play the same role in the quasilinear case as the Laplace and the heat operator play in the linear case). The authors show higher integrability of \(D_tu\) and \(D^2u\) and HĂ¶lder continuity of \(u\) or of \(Du\) or of \(D_tu\) and \(D^2u\) (depending on the space dimension \(n)\).

The book collects many techniques that are important in regularity theory (Miranda-Talenti estimates, Cacciopoli type estimates, Calderon-Zygmund kernels and their commutators). Hence, it will prove an essential reference for all those with an interest in solution regularity to elliptic and parabolic pdes’.

Reviewer: Lutz Recke (Berlin)

##### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

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

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

35B65 | Smoothness and regularity of solutions to PDEs |

35Jxx | Elliptic equations and elliptic systems |

35Kxx | Parabolic equations and parabolic systems |