Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds (Q2709431)

This monograph is devoted to the study of boundary value problems for second-order elliptic operators in Lipschitz subdomains of Riemannian manifolds. Already A. P. Calderon and A. Zygmund reduced in there early works boundary value problems of partial differential equations over smooth bounded domains in \(\mathbb{R}^n\) to pseudodifferential equations on their boundary with so-called boundary projections of Calderon's type. Calderon's projection plays nowadays an important role for the solution of problems of mathematical physic. For instance it is possible to decompose a total wave field in acoustics into its incoming and outgoing components with respect of geometric conditions. In 1962 independently S. G. Michlin and N. I. Muschelishvili introduced the notions of symbol, ellipticity, and regularizator into the theory of singular integral operators. The symbolic calculus for pseudodifferential operators is the work of Calderon and Zygmund (1957), Dynin (1961), Agranovic (1965), Hörmander (1965), Kohn and Nirenberg (1965) as well as Vishik and Eskin (1965). Firstly really remarkable results on the boundedness of singular integral operators on Lipschitz curves appear in the famous common work by A. P. Calderon, R. Coifman, A. McIntosh and Y. Meyer in 1982.NEWLINENEWLINENEWLINEIn order to attack new classes of boundary value problems of elliptic partial differential equations in domains with Lipschitz boundaries and in subdomains of Riemannian and Lipschitz manifolds new methods are required. In Chapter 1 a smooth, connected, compact, oriented, boundaryless manifold \(\Omega\) of real dimension \(m\) is introduced and equipped with a Lipschitzian Riemannian metric tensor \(g\). Over this manifold classes of functions and distributions with values in corresponding vector bundles \({\mathcal E}\) endowed with a Hermitian structure are introduced. On arbitrary Lipschitz subdomains of the manifold \({\mathcal M}\) are defined layer potential operators, which are discussed due to their boundedness in the corresponding \(L^p(\Omega, {\mathcal E})\) spaces and jump formulas. In Chapter 2 a strongly elliptic partial differential operator \(L\) is considered which acts on global sections of the bundle \({\mathcal E}\) into sections of the bundle \({\mathcal F}\). Under suitable assumptions it is shown that the Schwartz kernel of \(L^{-1}\) is invertible. As important example can be seen the Hodge-Laplacian operator on differential forms. In Chapter 3 the authors describe exactly the assumptions of the general second-order strongly elliptic system \(L(x,D)=L\). It is introduced the so-called nonsingularity hypothesis relatively to the Lipschitz domain \(\Omega\): NEWLINE\[NEWLINEu\in H_0^{1,2} (\Omega,{\mathcal E}),\;Lu=0\text{ in }\Omega\text{ then it follows }u=0 \text{ in }\Omega,NEWLINE\]NEWLINE which is for strongly elliptic operators fulfilled. By using double and single layer potentials global representation formulae for general elliptic systems are produced. Maybe the first use for nonregular bounded domains was done in 1962 by Yu. D. Burago, V. G. Maz'ya, V. D. Sapoznikova, and later in more general context by A. P. Calderon. Dirichlet's problem for the Hodge Laplacian is subject of consideration in Chapter 4. It is introduced the exterior derivative operator \(d\) and by the help of Christoffel symbols its formal adjoint \(\delta\). Then the Hodge Laplacian can be described as NEWLINE\[NEWLINE\Delta=-(d\delta +\delta d).NEWLINE\]NEWLINE A very general theorem on the Dirichlet problem for harmonic forms of arbitrary degree in Lipschitz domains on manifolds \((\Delta u=0)\) is proved. The treatment of a large class of further natural conditions for such harmonic forms follows in Chapter 5. This part seems to be very useful for a wide range of applications on boundary value problems of equations of mathematical physics. Chapter 7 is devoted the systematically development of boundary layer operators which are adapted to the Schrödinger operator \(\Delta -V\) \((V\geq 0)\). Rellich type estimates for differential forms which are defined on Lipschitz subdomains of \({\mathcal M}\) are discussed in Chapter 7. In Chapter 8 the Fredholmness of the operators \(\pm{1\over 2}I+M_\ell\), \(\pm {1 \over 2} I+N_\ell\) where \(M_\ell\) and \(N_\ell\) are corresponding boundary layer mappings. New kinds of boundary derivatives are introduced and discussed in Chapter 9. These operators are used to obtain a decomposition result for the Hilbert space \(L^2_tan(\partial \Omega,\Lambda^\ell T{\mathcal M})\) and \(L^2_n or (\partial\Omega, \Lambda^\ell T{\mathcal M})\). In continuation of results obtained in Chapter 8 in Chapter 10 the operators \(\lambda I+M_\ell\) and \(\lambda I+ N_\ell\) are proved as Fredholm operators with index zero for suitable \(\lambda\). In Chapters 11 and 14 is focussed the attention more on applications to \(\ell\)-harmonic fields with vanishing normal component on the boundary as well as to Maxwell's equations in Lipschitz domains. The Chapters 12 and 13 contain proofs of results formulated in former paragraphs.NEWLINENEWLINENEWLINEIn Appendix A a brief introduction is given for readers which are not enough familiar with the analysis on Lipschitz manifolds. This book is an essential contribution to the treatment of global boundary problems in nonsmooth Riemannian manifolds.