Pseudo-differential crack theory (Q2730962)

The authors develop a general theory of crack problems for elliptic partial differential and pseudodifferential equations. Namely, the crack is described by a surface \(S\) with boundary \(Y= \partial S\), embedded in a domain \(G\), where \(S\) is of codimension \(1\) in \(G\). Given a system \(A\) of operators in \(G\) and elliptic boundary conditions \(T_{\pm}\) on both sides \(S_\pm\) of \(S\), one studies the solutions of NEWLINE\[NEWLINEAu= f\quad\text{in }G-S,\quad T_{\pm}u= g_{\pm}\quad\text{on int }S_{\pm}.NEWLINE\]NEWLINE Locally, describing \(G-S\) by \((\mathbb{R}^2- \mathbb{R}_+)\times \Omega\) and passing to polar coordinates \((\rho,\varphi)\) in \(\mathbb{R}^2\setminus \{0\}\), we may write \(A\) in the form NEWLINE\[NEWLINEA= \rho^{-m} \sum_{k+ |\lambda|\leq m} a_{k\lambda}(\rho,y)\;(-\rho\partial/\partial\rho)^k(\rho D_y)^\lambdaNEWLINE\]NEWLINE with operator-valued coefficients, namely \(a_{k\lambda}(\rho, y)\) is for every \((\rho, y)\in \mathbb{R}_+\times \Omega\) a system of differential operators on the interval \(I= [0,2\pi]\) with coefficients smooth in \(\varphi\) up to \(\varphi= 0\) and \(\varphi= 2\pi\).NEWLINENEWLINENEWLINEParametrices are constructed for such problems, with extra trace and potential conditions along \(Y\), and the asymptotics of the solutions near \(Y\) is characterized. Applications are given, concerning in particular the second-order \(3\times 3\)-Lamé system in \(G\subset \mathbb{R}^3\) NEWLINE\[NEWLINE\mu\Delta u+ (\lambda+\mu)\text{ grad div }u= f,\quad \mu>0,\quad \lambda+\mu> 0NEWLINE\]NEWLINE with Dirichlet or Neumann boundary conditions on \(S_{\pm}\).