Higher Order Elliptic Equations on Nonsmooth Domains
Jun Geng
TL;DR
This work extends the Jerison–Kenig framework from second-order to higher-order elliptic systems with constant coefficients on bounded Lipschitz and convex domains, establishing uniform $W^{\ell,p}$ estimates and $L^p$ Dirichlet solvability in sharp, dimension- and order-dependent ranges. The authors develop two complementary proof strategies: a weak reverse Hölder/Good-$\lambda$ approach that yields $2<p<2+\frac{2}{d+2-\lambda}$ for suitable $\lambda$, and a duality-based method to handle $p<2$, yielding a comprehensive set of $p$-ranges for $\mathcal{L}(D)u=0$ and the polyharmonic equations. For the biharmonic case, they introduce a weighted positivity inequality on Lipschitz domains, enabling new WRH-type estimates and sharp ranges that extend beyond prior results, with convex domains admitting $L^p$ solvability for all $1<p<\infty$. The paper also proves the sharpness of these $p$-ranges via counterexamples based on cone exteriors and spherical eigenvalue problems, highlighting fundamental limits in non-smooth geometries. Collectively, these results advance the theory of higher-order boundary value problems on non-smooth domains and provide practical criteria for the solvability of Dirichlet problems in $L^p$ spaces.
Abstract
In 1995, D. Jerison and C. Kenig in \cite{JK-1995} considered the the inhomogeneous Dirichlet problem $Δu= f$ on $Ω$, $u=0$ on $\partialΩ$ in Lipschitz domains. One of their main results shows that the $W^{1,p}$ estimate holds for the sharp range $\frac{3}{2}-\varepsilon<p<3+\varepsilon$ for $d\geq 3$ and $\frac{4}{3}-\varepsilon<p<4+\varepsilon$ if $d=2$. Although the argument employed in \cite{JK-1995} yields optimal results, they rely on an essential fashion on the maximum principle and, as such, do not readily adapt to higher-order case. By using a new method, the aim of this paper is to establish an extension of their theorem for higher order inhomogeneous elliptic equations on bounded Lipschitz and convex domains, uniform $W^{\ell,p}$ estimates are obtained for $p$ in certain ranges. Especially, compare to the result in \cite{MM-2013} for biharmonic equation, a larger, sharp, range of $p's$ was obtained in this paper.