In the present article we study a minimization problem in \(\mathbb R^N\) involving the perimeter of the positivity set of the solution u and the integral of \(|\nabla u|^{p(x)}\). Here p(x) is a Lipschitz continuous function such that \(1<p_\mathrm{min}\le p(x)\le p_\mathrm{max}<\infty \). We prove that such a minimizing function exists and that it is a classical solution to a free boundary problem. In particular, the reduced free boundary is a \(C^2\) surface and the dimension of the singular set is at most \(N-8\). Under further regularity assumptions on the exponent p(x) we get more regularity of the free boundary. In particular, if \(p\in C^\infty \) we have that \(\partial _\mathrm{red}\{u>0\}\) is a \(C^\infty \) surface.

在本文中，我们研究\（\ mathbb R ^ N \）中的最小化问题，该问题涉及解u的正集合的周长和\（| \ nabla u | ^ {p（x）} \）的积分。这里p（x）是一个Lipschitz连续函数，使得\（1 <p_ \ mathrm {min} \ le p（x）\ le p_ \ mathrm {max} <\ infty \）。我们证明存在这样的最小化函数，并且它是自由边界问题的经典解决方案。特别地，减小的自由边界是\（C ^ 2 \）曲面，奇异集的维数最多为\（N-8 \）。在指数p（x），我们可以得到更多的自由边界规则性。特别是，如果\（p \ in C ^ \ infty \）我们有\（\ partial _ \ mathrm {red} \ {u> 0 \} \）是\（C ^ \ infty \）曲面。