Solutions of singular elliptic equations via the oblique derivative problem
The author studies a class of second order elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, \textit{M. Langlais} [Proc. Lond. Math. Soc., III. Ser. 50, 282--298 (1985; Zbl 0566.35052)] proved that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. The author proves some Schauder estimates for these equations even in Lipschitz domains. In addition, the author shows that bounded solutions of such problems are as smooth as the data allow. A key step to get these new results consists in noting that smooth solutions must satisfy an oblique derivative boundary condition and in applying the techniques developed by the author to study the oblique derivative problem in the case of non-degenerate elliptic equations.
scientific article; zbMATH DE number 5947316
scientific article