On some improperly posed problem for degenerate quasilinear elliptic equations (Q1335551)

The author gives a conditional stability estimate of a (regular) solution of the quasilinear (and possible degenerated) elliptic equation \[ \sum \partial_ j \bigl( | \partial ju |^{p-2} \partial_ ju) = 0 \] in a convex domain \(D \subset \mathbb{R}^ n\) in terms of the \(L_ p\)- norm of \(| u | + | \nabla u |\) on \(\Gamma \subset \partial D\). It implies that if \(u = \partial_ \nu u = 0\) on \(\Gamma\) then \(u = 0\) on the convex hull of \(D\). It does not however resolve the uniqueness question for the Cauchy problem for this equation because it is nonlinear and subtraction of the equations for two different solutions changes the form of the equations. The author mentions a counterexample of Martio showing that convexity assumption is important.