$L^2$-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 $2m$: \[ L\left ( x, \nabla \right )u = f\left ( x \right ) \qquad in \Omega \], \[ {\nabla ^k}u = 0 \qquad on \partial \Omega \left ( k = 0,...,m - 1 \right )\]. 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.

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R. Adams, *Sobolev Spaces*, Academic Press, New York, 1975
S. Agmon, *Lectures on Elliptic Boundary Value Problems*, Van Nostrand, Princeton, NJ, 1965
S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions 1, Comm. Pure Appl. Math. **12**, 623–727 (1959)
S. Agmon, A. Douglis, and L. Nirenberg, *Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions* 2, Comm. Pure Appl. Math. **17**, 35–92 (1964)
C. Bandle and M. Flucher, *Table of Inequalities in Elliptic Boundary Value Problems*, Math. Appl., Vol. 430, Kluwer, Dordrecht, 1998
C. Dafermos and W. Hrusa, *Energy methods for quasilinear hyperbolic initial-boundary value problems*, Arch. Rational Mech. Anal. **87**, 267–292 (1985)
S. Ebenfeld, *Aspekte der Kontinua mit Mikrostruktur*, Ph.D. thesis, Darmstadt, Shaker-Verlag, 1998
M. Giaquinta, *Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems*, Princeton University Press, 1983
D. Gilbarg and N. Trudinger, *Elliptic Partial Differential Equations of Second Order*, Springer, 1983
T. Kato, *Abstract Evolution Equations and Nonlinear Mixed Problems*, Lezioni Fermiane Pisa, 1988
H. Koch, *Hyperbolic Equations of Second Order*, Ph.D. thesis, Heidelberg, 1990
H. Koch, *Mixed problems for fully nonlinear hyperbolic problems*, Math. Zeit. **214**, 9–42 (1993)
A. Koshelev, *Regularity Problem for Quasilinear Elliptic and Parabolic Systems*, Springer-Verlag, New York, 1995
A. Milani, *A Regularity Result for Strongly Elliptic Systems*, Bollettino Un. Mat. Ital. B **2**, 641–651 (1983)
A. Milani, *A remark on the Sobolev regularity of classical solutions to strongly elliptic equations*, Math. Nachr. **190**, 203–219 (1998)
C. B. Morrey, *Multiple Integrals in the Calculus of Variations*, Springer, 1966
M. Renardy and R. Rogers, *An Introduction to Partial Differential Equations*, Springer-Verlag, New York, 1992
J. Wloka, *Partielle Differentialgleichungen*, Sobolevräume und Randwertaufgaben, Teubner, Stuttgart, 1982

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© Copyright 2002
American Mathematical Society