Rendiconti del Seminario Matematico della Università di Padova


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Volume 131, 2014, pp. 67–76
DOI: 10.4171/RSMUP/131-5

A product property of Sobolev spaces with application to elliptic estimates

Henry C. Simpson[1] and Scott J. Spector[2]

(1) Department of Mathematics, University of Tennessee, TN 37996-1300, Knoxville, USA
(2) Department of Mathematics, Southern Illinois University, IL 62901, Carbondale, USA

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for $p\in[1,\infty)$, $n,m\in\mathbb Z^+$, and $m\ge2$ that satisfy $m>n/p$, $$\left\Vert\phi\psi\right\Vert_{m,p,\Omega}\le K \left[ \left( \sup_{\Omega_s} \left\vert \phi \right\vert \right) \left\Vert\psi\right\Vert_{m,p,\Omega} + \Bigl(\left\Vert\psi\right\Vert_{m-1,q,\Omega}+\left\Vert\psi\right\Vert_{m-1,p,\Omega}\Bigr)\right] \left\Vert\phi\right\Vert_{m,p,\Omega} $$ for all $\phi,\psi \in W^{m,p}(\Omega)$ that satisfy $\rm {spt} \psi \subset \Omega_s\subset \Omega$ and domains $\Omega\subset\mathbb R^n$ that are nonempty, open, and satisfy the cone condition. Here $q=p$ if $p>n$, $q\in (n/\Upsilon,pn/(n-p)]$ if $n>p$, $q\in (n/\Upsilon,\infty)$ if $p=n$, $K=K(n,p,m,q,\mathcal C)$, where $\mathcal C$ is the cone from the cone condition, and $\Upsilon:= [[{n/p}]]$, the largest integer less than or equal to $n/p$.

Keywords: Elasticity, elliptic regularity, Sobolev estimate, systems of partial differential equations

Simpson Henry, Spector Scott: A product property of Sobolev spaces with application to elliptic estimates. Rend. Sem. Mat. Univ. Padova 131 (2014), 67-76. doi: 10.4171/RSMUP/131-5