اصل فوق العاده بر راه حل های ویسکوزیته معادلات لاپلاس بی نهایت Superposition principle on the viscosity solutions of infinity Laplace equations

1. Introduction The infinity Laplace equation △∞u(x) := ∑ 1≤i,j≤n uxi uxj uxixj = 0 was introduced by G. Aronsson [1] in the 1960s. R. Jensen [10] proved the equivalence of the infinity Laplace equation and the absolutely minimizing Lipschitz extension problem. He also proved the existence and uniqueness of the viscosity solution to the Dirichlet problem: △∞u = 0 in Ω, u = g on ∂Ω for any bounded domain Ω ⊂ R n and g ∈ C(∂Ω). Crandall–Evans–Gariepy [2] introduced the property of comparison with cones and proved that it is a characteristic property of infinity harmonic functions. The interior regularity for infinity harmonic functions was achieved by Evans, Savin and Smart in [4,14] and [3]. The boundary regularity was studied by Wang–Yu [15], Hong [6,8] and Hong-Liu [9]. The inhomogeneous infinity Laplace equation: △∞u = f in Ω (1) was introduced by Lu-Wang [13]. Lindgren [11] proved that the blow-ups are linear if f ∈ C(Ω)∩L∞(Ω) and u is everywhere differentiable if f ∈ C 1 (Ω) ∩ L∞(Ω). Hong [7] proved the boundary differentiability of u at a differentiable boundary point and Feng–Hong [5] studied the slope estimate and boundary differentiability of u on the convex domains. In [11], Lindgren constructed an extension u˜(x1, . . . , xn+2) = u(x1, . . . , xn) + 5xn+1 + C|xn+2| 4 3 and used the conclusion that if △∞u = f in R n in the viscosity sense then △∞u˜ = f + 2 6 3 4 C in R n+2 in the viscosity sense without a proof. Both of the papers [7] and [5] used the same extension and conclusion. The purpose of the extension is to make the slope function strictly positive and the inhomogeneous term bounded away from 0. The conclusion seems obvious but we will see it is not so. In the book [12](Page 58), Lindqvist also used the similar extension and conclusion. The author gave a very short proof of the conclusion in the footnote, but we do not think the proof is strict enough. The last sentence of the proof says “The desired inequality follows”, but we cannot see why the inequality follows from the proceeding deduction. The argument does not involve an analysis on the second order derivatives, the counterexample in this paper indicates that one should not prove the conclusion without going deep into the analysis on the second order derivatives. In this note, we will give a strict proof of the above mentioned conclusion and provide a counterexample to show that the things are not that simple. We begin by recalling the definition of viscosity solution.