On the best Hölder exponents for generalized solutions of the Dirichlet problem for a second order elliptic equation. (English. Russian original) Zbl 0633.35020

Math. USSR, Sb. 59, No. 1, 113-127 (1988); translation from Mat. Sb., Nov. Ser. 131(173), No. 1, 113-125 (1986).
The behaviour of the generalized solutions of the Dirichlet problem for an elliptic second order equation in a boundary point of the domain \(\Omega\) is considered in the paper. Under certain assumptions on the structure and smoothness of the boundary of \(\Omega\) in the neighbourhood of the boundary point and the smoothness of the coefficients of the equation for the generalized solution of the Dirichlet problem reducing to zero on the boundary, a power module of continuity is set at the given point. An example illustrating that this index is unimprovable, but the assumptions are essential, is given.
Using the results of the module of continuity, it is proved that the generalized solution in the domain considered belongs to the Hölder space \(C^{\gamma}\) in the isolated domain. Index \(\gamma\) is determined by the structure of the domain boundary and is unimproved in the given class of domains.
Reviewer: B.Kvedaras


35J25 Boundary value problems for second-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
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