Estimates for elliptic systems from composite material.

*(English)*Zbl 1125.35339Suppose 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.

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.

Reviewer: Guido Sweers (MR1990481)

##### MSC:

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

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\textit{Y. Li} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 56, No. 7, 892--925 (2003; Zbl 1125.35339)

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