Estimates for elliptic systems from composite material. (English) Zbl 1125.35339

Suppose that \(D\) is a bounded domain in \(\mathbb R^n\) that contains subdomains \(D_m\), \(m=1,\dots ,L\), with \(D=\bigcup D_m\). Let \(\sum_{\alpha ,\beta =1}^{n}\sum_{j=1}^{N}\partial _{\alpha }A_{i,j}^{\alpha ,\beta }\partial _{\beta }u_{j}=b_{i}\) with \(i=1,\dots ,N\) be a (weakly) elliptic system of equations where \(A_{i,j}^{\alpha ,\beta }\) are Hölder continuous in \( D_m\) but not necessarily continuous on \(\partial D_m\). Such problems appear naturally in elasticity theory of composite material. The version of the elliptic condition used here indeed does allow for these systems. The estimates that Li and Nirenberg seek to establish answer the following question: does an interior Hölder type bound exist for \(\nabla u\) in terms of \(u\) and \(b?\) Assuming that the boundaries \(\partial D_m\) are \( C^{1,\gamma }\) their main result gives the affirmative answer.
Set \( D_{\varepsilon }=\left\{x\in D; \text{dist}\left( x,\partial D\right) >\varepsilon \right\}\). Then there exist \(C\) and \(\gamma ^*>0\) such that for all \( \gamma ^{\prime }\in \left( 0,\gamma ^{*}\right) \) and \( b_{i}:= h_i+\sum_{\beta =1}^n\partial _{\beta }g_i^{\beta}\) any weak solution \(u\) satisfies \[ \sum_{m=1}^{L}\left\| u\right\| _{C^{1,\gamma ^{\prime }}(\overline{D} _{m}\cap D_{\varepsilon })}\leq C\left( \left\| u\right\| _{L^{2}(D)}+\left\| h\right\| _{L^{\infty }(D)}+\sum_{m=1}^{L}\left\| g\right\| _{C^{\gamma ^{\prime }}(\overline{D}_m)}\right). \] The constant \(C\) that is obtained does not depend on the distance between subdomains \(D_m\) and hence allows even some “touching” of subdomains. Although related results are available in the literature, the present combination of “system” and “composite material” is new and makes the long and hard analysis in this paper necessary. The authors mention a preceding result for the scalar equation which is due to Li and Vogelius. The present paper also pays tribute to results of Chipot, Kinderlehrer, and Vergara-Caffarelli, and of Avellaneda and Lin.


35J55 Systems of elliptic equations, boundary value problems (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
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[1] Avellaneda, Comm Pure Appl Math 40 pp 803– (1987)
[2] Babur?ska, Comput Methods Appl Mech Engrg 172 pp 27– (1999)
[3] ; ; Asymptotic analysis for periodic structures. Studies in Mathematics and Its Applications, 5. North-Holland, Amsterdam-New York, 1978.
[4] Bonnetier, SIAM J Math Anal 31 pp 651– (2000)
[5] Caffarelli, Ann of Math (2) 130 pp 189– (1989)
[6] ; Fully nonlinear elliptic equations. American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, R.I., 1995. · Zbl 0834.35002
[7] Caffarelli, Comm Pure Appl Math 51 pp 1– (1998)
[8] Chipot, Arch Rational Mech Anal 96 pp 81– (1986)
[9] Multiple integrals in the calculus of variations and nonlinear elliptic systems. Annals of Mathematics Studies, 105. Princeton University, Princeton, N.J., 1983.
[10] Li, Arch Rational Mech Anal 153 pp 91– (2000)
[11] ; ; Mathematical problems in elasticity and homogenization. Studies in Mathematics and Its Applications, 26. North-Holland, Amsterdam, 1992. · Zbl 0768.73003
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