arXiv subjects

Joel Kilty

Publications and source records attributed to Joel Kilty.

3 recordsLinked to original sources

arXiv2009

A Bilinear Estimate for Biharmonic Functions in Lipschitz Domains

We show that a bilinear estimate for biharmonic functions in a Lipschitz domain $Ω$is equivalent to the solvability of the Dirichlet problem for the biharmonic equationin $Ω$. As a result, we prove that for any given bounded Lipschitz domain $Ω$ in $\rn{d}$ and $1<q<\infty$, the solvability of the $L^{q}$ Dirichlet problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{1,q}(\partialΩ)$ is equivalent to that of the $L^p$ regularity problem for $Δ^2 u=0$ in $Ω$ with boundary data in ${\emph{WA}}^{2,p}(\partialΩ)$, where $\frac{1}{p} +\frac{1}{q}=1$. This duality relation, together with known results on the Dirichlet problem, allows us to solve the $L^p$ regularity problemfor $d\ge 4$ and $p$ in certain ranges.

math.AP↗

arXiv2009

The $L^p$ Dirichlet Problem for the Stokes System on Lipschitz Domains

We study the $L^p$ Dirichlet problem for the Stokes system on Lipschitz domains. For any fixed $p>2$, we show that a reverse Hölder condition with exponent $p$ is sufficient for the solvability of the Dirichlet problem with boundary data in $L^p_N(\partialΩ,\rn{d})$. Then we obtain a much simpler condition which implies the reverse Hölder condition. Finally, we establish the solvability ofthe $L^p$ Dirichlet problem for $d\geq 4$ and $2-\varepsilon<p<\frac{2(d-1)}{d-3}+\varepsilon$.

math.AP↗

arXiv2009

The $L^p$ Regularity Problem on Lipschitz Domains

This paper contains two results on the $L^p$ regularity problem on Lipschitz domains. For second order elliptic systems and $1 2$, the solvability of the $L^p$ regularity problem is equivalent to a weak reverse Hölder condition with exponent $p$.

math.AP↗