Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
The Neumann, Dirichlet and regularity problems are studied in \(L^2(\mathbb R^n)\) for divergence form second order elliptic equations \(\text{div}_{t,x}A(x)\nabla_{t,x}U(t,x)=0\) on the half-space \(\mathbb R_+^{n+1}=\{ (t,x)\in \mathbb R\times \mathbb R^n; t>0\} \). Let \(A_0(x)=((a_0)_{ij}(x))\) (\(i,j=0,\dots ,n\)) be a \(t\)-independent complex, accretive coefficient matrix function with \((a_0)_{ij}\in L^\infty (\mathbb R^n)\). Furthermore assume that \(A_0(x)\) is either real symmetric of block form or constant. Then there exists \(\epsilon >0\) depending only on \(\| A\| _\infty \), the accretivity constant and the dimension \(n\), such that if \(A\in L_\infty (\mathbb R^n; L(C^{n+1}))\) is \(t\)-independent and satisfies \(\| A-A_0 \| _\infty <\epsilon \), then Neumann, Dirichlet and regularity problems are well posed in \(L_2\). The problems are studied by a new boundary operator method. It is also proved that certain transmission problems are well posed.
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