Boundary value problems on product domains
Dariush Ehsani
TL;DR
This work analyzes the inhomogeneous Dirichlet problem on product domains and derives an explicit asymptotic expansion of the solution up to the boundary, revealing the exact boundary singularities by constructing singular functions that compose the expansion. It extends Eh07's corollary for the $ar{\partial}$-Neumann problem to Dirichlet problems, building a systematic singular-function framework in which successive terms gain differentiability near the boundary. Locally, the boundary is flattened to a product of half-spaces using $y_j = x_j - \phi_j$, leading to a transformed Dirichlet problem for the operator $\triangle'$ on $\mathbb{H}^n_q$ with metric $g^{ij}$, and solved via odd/even reflections and Fourier analysis to produce a parametrix-like expansion $v^o = \sum_{j=0}^N v_j + v_{R_N}$ with a recurrence for the $v_j$. The explicit construction expresses the near-boundary solution as $u = \sum_{|k| \ge 0,\; l \ge 1,\; 2 \le \ell(p) \le q} c_{klp} \ \Phi_{lk}^p$ plus a smooth remainder, where the singular functions $\Phi_l^q$ and $\Phi_{lk}^q$ capture the leading singularities, and inverses Fourier transforming the leading terms shows the singular content is governed by transforms like $1/\eta^k$ and $(\sum a_i\eta_i^2)^{-l}$. Consequently, a near-boundary expansion $u = \sum c_{klp} \ \Phi_{lk}^p$ with a smooth remainder is obtained, and coefficients can be nonzero even for $f \equiv 1$.
Abstract
We consider the inhomogeneous Dirichlet problem on product domains. The main result is the asymptotic expansion of the solution in terms of increasing smoothness up to the boundary. In particular, we show the exact nature of the singularities of the solution at singularities of the boundary by constructing singular functions which make up an asymptotic expansion of the solution.