On steady boundaries under the scheme of viscosity solutions
The possible boundary values of viscosity solutions on removable boundaries are studied. On such boundaries the values of a solution are in general determined by the boundary conditions on the other part of the boundary and there is little information on those values. This work is concerned with the situation where the values depend only on the removable boundary itself, such removable boundaries will be called steady boundaries. Removable boundaries are related to degeneracy of elliptic equations. The main result of this paper shows that, in some assumptions, for any subsolution \(u\in USC(\overline\Omega)\) and supersolution \(\nu\in LSC(\overline \Omega)\), \[ u_b(x)= \limsup_{\Omega \ni y\to x} u(y) \quad \text{and} \quad v_b(x) =\liminf_{\Omega \ni y\to x} \nu(y) \] are subsolutions, resp. supersolutions of \(F(x,w(x), Dw(x), D^2w(x)) =0\) in \(B_r(\widehat x) \cap\partial \Omega\). Applications of the main theorem on studying asymptotic behavior are given in the last section.
scientific article; zbMATH DE number 1067764