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

In this article, we consider the following Dirichlet system of order $2m$: \[ L\left ( x, \nabla \right )u = f\left ( x \right ) \qquad in \Omega \], \[ {\nabla ^k}u = 0 \qquad on \partial \Omega \left ( k = 0,...,m - 1 \right )\]. 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: \[ {\left \| u \right \|_{{M^{m + s}}\left ( \Omega , {\mathbb {R}^N} \right )}} \le \hat C{\left \| f \right \|_{{H^{ - m + s}}\left ( \Omega ,{\mathbb {R}^N} \right )}} + \hat K{\left \| u \right \|_{{L^2}\left ( \Omega ,{\mathbb {R}^N} \right )}}\]. 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.