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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  • [1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR 0450957
  • [2] Shmuel Agmon, Lectures on elliptic boundary value problems, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0178246
  • [3] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 0125307, https://doi.org/10.1002/cpa.3160120405
  • [4] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 0162050, https://doi.org/10.1002/cpa.3160170104
  • [5] C. Bandle and M. Flucher, Table of inequalities in elliptic boundary value problems, Recent progress in inequalities (Niš, 1996) Math. Appl., vol. 430, Kluwer Acad. Publ., Dordrecht, 1998, pp. 97–125. MR 1609927
  • [6] Constantine M. Dafermos and William J. Hrusa, Energy methods for quasilinear hyperbolic initial-boundary value problems. Applications to elastodynamics, Arch. Rational Mech. Anal. 87 (1985), no. 3, 267–292. MR 768069, https://doi.org/10.1007/BF00250727
  • [7] S. Ebenfeld, Aspekte der Kontinua mit Mikrostruktur, Ph.D. thesis, Darmstadt, Shaker-Verlag, 1998
  • [8] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • [9] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [10] T. Kato, Abstract Evolution Equations and Nonlinear Mixed Problems, Lezioni Fermiane Pisa, 1988
  • [11] H. Koch, Hyperbolic Equations of Second Order, Ph.D. thesis, Heidelberg, 1990
  • [12] Herbert Koch, Mixed problems for fully nonlinear hyperbolic equations, Math. Z. 214 (1993), no. 1, 9–42. MR 1234595, https://doi.org/10.1007/BF02572388
  • [13] Alexander Koshelev, Regularity problem for quasilinear elliptic and parabolic systems, Lecture Notes in Mathematics, vol. 1614, Springer-Verlag, Berlin, 1995. MR 1442954
  • [14] Albert J. Milani, A regularity result for strongly elliptic systems, Boll. Un. Mat. Ital. B (6) 2 (1983), no. 2, 641–651 (English, with Italian summary). MR 716753
  • [15] Albert Milani, A remark on the Sobolev regularity of classical solutions of strongly elliptic equations, Math. Nachr. 190 (1998), 203–219. MR 1611620, https://doi.org/10.1002/mana.19981900111
  • [16] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, 1966
  • [17] Michael Renardy and Robert C. Rogers, An introduction to partial differential equations, 2nd ed., Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004. MR 2028503
  • [18] Joseph Wloka, Partielle Differentialgleichungen, B. G. Teubner, Stuttgart, 1982 (German). Sobolevräume und Randwertaufgaben. [Sobolev spaces and boundary value problems]; Mathematische Leitfäden. [Mathematical Textbooks]. MR 652934

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


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