In this work we study a minimization problem with two-phases where in each phase region the problem is ruled by a quasi-linear elliptic operator of p−Laplacian type. The problem in its variational form is as follows: min v  ∫ Ω∩{v>0} ( 1 p |∇v|p +λ p +(x)+ f+(x)v ) dx+ ∫ Ω∩{v≤0} ( 1 q |∇v|q +λ q −(x)+ f−(x)v ) dx  . Here we minimize among all admissible functions v in an appropriate Sobolev space with a prescribed boundary datum v = g on ∂Ω. First, we show existence of a minimizer, prove some properties, and provide an example for non-uniqueness. Moreover, we analyze the limit case where p and q go to infinity, obtaining a limiting free boundary problem governed by the ∞−Laplacian operator. Consequently, Lipschitz regularity for any limiting solution is obtained. Finally, we establish some weak geometric properties for solutions.

中文翻译：

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$p$-Laplacians 驱动的非各向同性两相最小化问题的极限情况

在这项工作中，我们研究了一个具有两相的最小化问题，其中在每个相区域中，问题由 p-Laplacian 类型的准线性椭圆算子控制。变分形式的问题如下： min v  ∫ Ω∩{v>0} ( 1 p |∇v|p +λ p +(x)+ f+(x)v ) dx+ ∫ Ω∩ {v≤0} ( 1 q |∇v|q +λ q −(x)+ f−(x)v ) dx  。在这里，我们在具有指定边界数据 v = g on ∂Ω 的适当 Sobolev 空间中最小化所有可容许函数 v。首先，我们展示了最小化器的存在，证明了一些性质，并提供了一个非唯一性的例子。此外，我们分析了 p 和 q 趋于无穷大的极限情况，获得了由 ∞−Laplacian 算子控制的极限自由边界问题。因此，获得了任何极限解的 Lipschitz 正则性。最后，我们为解建立了一些弱几何性质。