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The Dirichlet problem with incompatible degeneration of initial data. (English. Russian original) Zbl 0843.35025
Russ. Acad. Sci., Dokl., Math. 50, No. 1, 104-107 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 337, No. 4, 447-449 (1994).
The Dirichlet problem is studied for a second-order elliptic equation with incompatible degeneration of initial data in an arbitrary convex domain \(\Omega\); the existence and uniqueness of an \(R_\nu\)-generalized solution in the weighted space \(W^1_{2, \nu+ \beta/2}(\Omega^*)\) are established, where \(\Omega^*\) is a subdomain of \(\Omega\) on which a certain condition on the coefficients holds. The existence of an \(R_\nu\)-generalized solution in \(W^1_{2, \nu+ \beta/2}(\Omega)\) is proved, and estimates of the solution are obtained in terms of the norm in \({\mathcal L}_{2, \mu}\) of the right-side of the equation.
35J15 Second-order elliptic equations
35J70 Degenerate elliptic equations