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On absolutely minimizing Lipschitz extensions and PDE \(\bigtriangleup_{\infty}(u)=0\). (English) Zbl 1154.35055

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.

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
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