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Asymptotic representation of solutions to the Dirichlet problem for elliptic systems with discontinuous coefficients near the boundary. (English) Zbl 1220.35013

The paper is concerned with the variational solutions to the Dirichlet problem in \(\mathbb R_{+}^n \backslash \{O\}\) and in \(B_{+}(\delta)=\mathbb R_{+}^n\cap B(\delta)\), for the matrix differential operator \(\mathcal{L}(x, D_x)=L(D_x) - N(x, D_x)\). Here \(L(D_x)=\sum_{| \alpha| =| \beta| =m}L_{\alpha \beta} D_x^{\alpha+\beta}\) is a operator with constant coefficients, \(N(x, D_x) = \sum_{| \alpha| ,| \beta| \leq m}D_x^{\alpha}(N_{\alpha \beta}(x)D_x^{\beta}u)\) is treated as a perturbation operator and is characterized by the function \(\kappa(x) = \sum_{| \alpha| ,| \beta| \leq m} x_n^{2m-| \alpha+\beta| }| N_{\alpha \beta}(x)| \). Under the ellipticity and a local estimate outside the origin of \(\mathcal L(x, D_x)\), the boundedness of \(\kappa\) and the smalless of \(\kappa_s(r)=\big(\int_{ r/e<| y| <r}\kappa^s(y)| y| ^{-n}dy\big)^{ 1/s}\) for some \(s\geq 1\), the author describes the structure of these solutions near the origin and derives the explicit sharp estimates for them. The author also gives several corollaries of the main results. In particular, in the scalar case, this work improves the asymptotic formulae from the article by the author and V. Mazya [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 2, No. 3, 551–600 (2003; Zbl 1170.35340)].
Reviewer: Anh Cung (Hanoi)

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35C20 Asymptotic expansions of solutions to PDEs
35D40 Viscosity solutions to PDEs

Citations:

Zbl 1170.35340
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