Le Gruyer, E. On absolutely minimizing Lipschitz extensions and PDE \(\bigtriangleup_{\infty}(u)=0\). (English) Zbl 1154.35055 NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 1-2, 29-55 (2007). The author shows the existence of absolutely minimizing Lipschitz extensions assuming Jensen’s hypotheses and using a new method based on numerical schemes for the viscosity solution of the equation \(\Delta_{\infty}( u) =0,\) under Dirichlet condition.Next, the maximum principle is established for the functional equation \(u( x) =\mu( u;x)\), \(\forall x\in G-S\), under the condition \(u( s) =f( s) \), \(\forall s\in S\). As a consequence, one obtains the uniqueness of the solution. The existence and the stability are also proved. At the end, the author prove the existence of an absolutely minimizing Lipschitz extension as a limit of solutions of the above problem. Reviewer: N. C. Apreutesei (Iaşi) Cited in 25 Documents MSC: 35J70 Degenerate elliptic equations 35B50 Maximum principles in context of PDEs 35B35 Stability in context of PDEs 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs Keywords:infinity Laplacian; harmonious extensions; maximum principles; viscosity solutions PDFBibTeX XMLCite \textit{E. Le Gruyer}, NoDEA, Nonlinear Differ. Equ. Appl. 14, No. 1--2, 29--55 (2007; Zbl 1154.35055) Full Text: DOI