On quasilinear elliptic equations in domains with conical boundary points. (English) Zbl 0655.35022

Dobrowolski, Manfred

J. Reine Angew. Math. 394, 186-195 (1989).

For the solution u of the variational problem \[ \int_{\Omega} \{1/p| \nabla u| p-fu\}dx \to \text{Min in } \overset\circ H^{1,p}(\Omega),\quad \Omega\subset\mathbb{R}^ n, 1<p<\infty, \] it is shown that \(| u| \leq cr^{\alpha}\) near a conical boundary point. Here, \(\alpha\) is given by the solution \((\alpha,t(\sigma))\) of a nonlinear eigenvalue problem which does not depend on \(f\). For a corner with interior angle \(\omega\), the eigenvalue \(\alpha\) can be determined exactly. In the important case of a slit domain, i.e. \(\omega=2\pi\), we obtain \(\alpha = (p-1)/p.\) Moreover, for \(f\geq 0\) it is proved that the solution u can be expanded in the form \(ks+w\) with \(k\in\mathbb{R}\), \(s=r^\alpha t(\sigma)\) and \(| w| \leq cr^{\alpha+\eta}\). These results also extend to more general variational problems \[ \int_\Omega \{a(| \nabla u|) - fu\} dx \to \text{Min with } a(| \nabla u|) \sim | \nabla u|^ p. \]

Reviewer: M.Dobrowolski

MSC:

35J60	Nonlinear elliptic equations
49J20	Existence theories for optimal control problems involving partial differential equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B45	A priori estimates in context of PDEs

Keywords:

quasilinear; conical boundary points; variational problem; nonlinear eigenvalue problem; corner