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Media type:
E-Article
Title:
L²-REGULARITY THEORY OF LINEAR STRONGLY ELLIPTIC DIRICHLET SYSTEMS OF ORDER 2m WITH MINIMAL REGULARITY IN THE COEFFICIENTS
Contributor:
EBENFELD, STEFAN
imprint:
Brown University, 2002
Published in:Quarterly of Applied Mathematics
Language:
English
ISSN:
0033-569X;
1552-4485
Origination:
Footnote:
Description:
<p>In this article, we consider the following Dirichlet system of order 2m: L(x, ∇)u = f(x) in Ω, ∇ku = 0 on ∂Ω (k = 0, . . . , m - 1). Here, Ω is a smooth bounded domain in Rn and the differential operator L(x, ∇) 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 ∊ H-m+s(Ω, RN), every weak solution $u \in H_0^m\left( {\Omega ,{R^N}} \right)$ is actually in Hm+s(Ω, RN) and satisfies an a priori estimate of the following form: $\parallel u{\parallel _{{M^{m + s}}\left( {\Omega ,{R^N}} \right)}} \leqslant \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{C} \parallel f{\parallel _{{H^{ - m + s}}\left( {\Omega ,{R^N}} \right)}} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{K} \parallel u{\parallel _{{L^2}\left( {\Omega ,{R^N}} \right)}}$ 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(x, ∇) 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. Here, we prove the above regularity result under the assumptions (2), (8) on the coefficients and we give an explicit representation formula for the regularity constants Ĉ and K̂ (see (10)).</p>