The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge.
The author investigates the behaviour of solutions to boundary value problems (Dirichlet, Neumann, mixed) for the equation \[ -\frac{d}{dx_i}a_i(x,u,u_x)+a(x,u,u_x)+b(x,u,u_x)=f(x) \] in a domain \(G\subset \mathbb R^n\), \(n\geq 3\). The coefficients \(a_i(x,u,u_x)\) satisfy a degenerate ellipticity condition: a typical example is \[ a_i(x,u,u_x)= (x^2_{n-1}+x^2_n)^\frac{\tau}{2}| u| ^q| \nabla u| ^{m-2}u_{x_i}, \] \(q\geq 0\), \(m\geq 2\), \(\tau\geq m-2\). The functions \(a(x,u,u_x)\), \(b(x,u,u_x)\) and \(f(x)\) satisfy suitable restrictions. The domain \(G\) possesses an edge, and the paper is devoted mainly to the estimate of the weak solutions to the boundary value problems near this edge. Suitable weighted Sobolev spaces are introduced and an almost sharp estimate of \(u(x)\) in a neighborhood of the edge is given.
degenerate quasilinear elliptic equations
scientific article; zbMATH DE number 2056129
scientific article