Global Hölder solvability of second order elliptic equations with locally integrable lower-order coefficients
Takanobu Hara
TL;DR
This work establishes global Hölder solvability for the Dirichlet problem $-\mathrm{div}(A\nabla u) + \mathbf{b}\cdot\nabla u + \mu u = \nu$ in a bounded domain $\Omega$, with boundary data $u=g$, under a capacity density condition on the boundary and Morrey-type control of the lower-order terms. The authors introduce Morrey-type spaces $\mathsf{M}^{\lambda}(\Omega)$ and use an Ancona-style, Green-function-free approach by formulating a compact perturbation operator $T$ and applying the Fredholm alternative to obtain existence, uniqueness, and Hölder regularity of the solution. The main result requires $|\mathbf{b}|^{2} m \in \mathsf{M}^{n-2+2\beta}(\Omega)$, $\mu \in \mathsf{M}^{n-2+\beta}(\Omega)$ with $\mu\ge 0$, and $\nu, g$ in corresponding Morrey-type data, yielding a bound $\|u\|_{C^{\beta_{*}}(\overline{\Omega})}$ controlled by data and domain diameter. This framework broadens global boundary regularity results to locally integrable lower-order terms, without energy-coercivity assumptions, and strengthens the understanding of boundary behavior in elliptic problems with rough coefficients.
Abstract
We prove the existence of globally Hölder continuous solutions to certain elliptic partial differential equations with lower-order terms. Our result is applicable to coefficients controlled by a negative power of the distance from the boundary of the domain and significantly improves Theorem 8.30 in Gilbarg and Trudinger (1983). The proof is derived by applying the strategy of Ancona (1986) to a new Morrey-type space.