Asymptotics for elliptic mixed boundary problems. Pseudo-differential and Mellin operators in spaces with conormal singularity.

*(English)*Zbl 0689.35104
Mathematical Research, 50. Berlin: Akademie-Verlag. 418 p. DM 42.00 (1989).

This monograph presents a theory of pseudo-differential boundary value problems, with and without the transmission property, on manifolds with conical singularities and with edges. The basic idea is to study appropriate algebras of operators which contain in particular the parametrices of boundary problems for differential operators. The first two chapters are devoted to the calculus on \(R_+\). Spaces of functions \(f(t),t>0\), with suitable asymptotic expansions as \(t\to 0\) and \(t\to \infty\), and spaces of Sobolev type are introduced.

Operators on these spaces are also studied: Green operators - or smoothing operators, Mellin operators and pseudo-differential operators; the symbolic calculus is based on the Mellin formulation. These operators generate a minimal algebra which is closed under parametrix constructions. In order to prepare the theory on \(R^ n_+=R^{n- 1}\times R_+,\) families of operators on \(R_+\) depending on a parameter \(x'\in R^{n-1}\) are also considered. Chapter 3 deals with boundary value problems on a smooth manifold X with boundary \(\partial X\). The appropriate function spaces as well as Green operators, Mellin operators and pseudo-differential operators are first introduced in the local model \(R^ n_+\). Then a class of boundary value problems for operators acting between sections of vector bundles over X and \(\partial X\) is investigated. For elliptic problems, Fredholm property, existence of parametric, regularity and asymptotic properties of solutions are proved. In chapter 4, similar results for mixed boundary value problems on manifolds with conical singularities and with edges are given.

Operators on these spaces are also studied: Green operators - or smoothing operators, Mellin operators and pseudo-differential operators; the symbolic calculus is based on the Mellin formulation. These operators generate a minimal algebra which is closed under parametrix constructions. In order to prepare the theory on \(R^ n_+=R^{n- 1}\times R_+,\) families of operators on \(R_+\) depending on a parameter \(x'\in R^{n-1}\) are also considered. Chapter 3 deals with boundary value problems on a smooth manifold X with boundary \(\partial X\). The appropriate function spaces as well as Green operators, Mellin operators and pseudo-differential operators are first introduced in the local model \(R^ n_+\). Then a class of boundary value problems for operators acting between sections of vector bundles over X and \(\partial X\) is investigated. For elliptic problems, Fredholm property, existence of parametric, regularity and asymptotic properties of solutions are proved. In chapter 4, similar results for mixed boundary value problems on manifolds with conical singularities and with edges are given.

Reviewer: P.Jeanquartier

##### MSC:

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

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

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

35J15 | Second-order elliptic equations |

35B40 | Asymptotic behavior of solutions to PDEs |

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