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This article is cited in 51 scientific papers (total in 51 papers)
On the Dirichlet problem for a second-order elliptic equation
A. K. Gushchin
Abstract:
The function space $C_{n-1}(\overline Q)$, $C(\overline Q)\subset C_{n-1}(\overline Q)\subset L_2(Q)$, where $Q$ is a bounded domain in $\mathbf R_n$, consists of elements that on sets of positive $(n-1)$-dimensional Hausdorff measure have traces with a property analogous to joint continuity. For $\partial Q\in C^1$ the set of traces of the functions in $C_{n-1}(\overline Q)$ on $\partial Q$ coincides with $L_2(\partial Q)$, and the imbedding $W_2^1(Q)\subset C_{n-1}(\overline Q)$ is valid.
Solutions of the Dirichlet problem in $C_{n-1}(\overline Q)$ are considered for the elliptic equation
$$
\sum_{i,j=1}^n(a_{ij}(x)u_{x_i})_{x_j}=f,\quad x\in Q;\qquad u|_{\partial Q}=u_0.
$$
Under the assumption that the normal to $\partial Q$ and the coefficients of the equation satisfy the Dini condition on $\partial Q$, it is established that for all $u_0\in L_2(\partial Q)$ and $f\in W_2^{-1}(Q)$ there is a unique solution that depends continuously on $u_0$ and $f$. It is proved that in this situation the solution in $C_{n-1}(\overline Q)$ coincides with the concept of a solution in $W^1_{2,\mathrm{loc}}$ introduced by Mikhailov.
Bibliography: 39 titles.
Received: 07.12.1987
Citation:
A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Mat. Sb. (N.S.), 137(179):1(9) (1988), 19–64; Math. USSR-Sb., 65:1 (1990), 19–66
Linking options:
https://www.mathnet.ru/eng/sm1764https://doi.org/10.1070/SM1990v065n01ABEH002075 https://www.mathnet.ru/eng/sm/v179/i1/p19
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Abstract page: | 838 | Russian version PDF: | 281 | English version PDF: | 18 | References: | 74 | First page: | 2 |
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