Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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$ L^2$-regularity theory of linear strongly elliptic Dirichlet systems of order $ 2m$ with minimal regularity in the coefficients


Author: Stefan Ebenfeld
Journal: Quart. Appl. Math. 60 (2002), 547-576
MSC: Primary 35J55; Secondary 35B45, 35B65
DOI: https://doi.org/10.1090/qam/1914441
MathSciNet review: MR1914441
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Abstract: In this article, we consider the following Dirichlet system of order $ 2m$:

$\displaystyle L\left( x, \nabla \right)u = f\left( x \right) \qquad in \Omega $

,

$\displaystyle {\nabla ^k}u = 0 \qquad on \partial \Omega \left( k = 0,...,m - 1 \right)$

. Here, $ \Omega $ is a smooth bounded domain in $ {\mathbb{R}^n}$ and the differential operator $ L\left( x, \nabla \right)$ given by (1) satisfies the Legendre-Hadamard condition (4). From the general elliptic theory we know that for sufficiently smooth coefficients $ A_{\alpha \beta }^{\left( m \right)}, B_{\alpha \beta }^{\left( {km} \right)}, C_\alpha ^{\left( k \right)}$ and for $ f \in \\ {H^{ - m + s}}\left( \Omega , {\mathbb{R}^N} \right)$, every weak solution $ u \in H_{0}^{m}\left( \Omega , {\mathbb{R}^N} \right)$ is actually in $ {H^{m + s}}\left( \Omega , {\mathbb{R}^N} \right)$ and satisfies an a priori estimate of the following form:

$\displaystyle {\left\Vert u \right\Vert _{{M^{m + s}}\left( \Omega , {\mathbb{R... ... \hat K{\left\Vert u \right\Vert _{{L^2}\left( \Omega ,{\mathbb{R}^N} \right)}}$

. The latter a priori estimate is of particular interest in applications to nonlinear PDEs (see, e.g., [6] and [10]). There the coefficients of $ L\left( x, \nabla \right)$ result from a linearization procedure and consequently they cannot be chosen as smooth as one likes. Therefore, e.g. in [10] (Kato), the author cannot use the famous results stated in [4] (Agmon-Douglis-Nirenberg) but refers to [14] (Milani) instead.

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DOI: https://doi.org/10.1090/qam/1914441
Article copyright: © Copyright 2002 American Mathematical Society

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