An optimization problem with volume constraint involving the \(\varPhi \)-Laplacian in Orlicz–Sobolev spaces is considered for the case where \(\varPhi \) does not satisfy the natural condition introduced by Lieberman. A minimizer \(u_\varPhi \) having non-degeneracy at the free boundary is proved to exist and some important consequences are established like the Lipschitz regularity of \(u_ \varPhi \) along the free boundary, that the set \(\{u_\varPhi >0\}\) has uniform positive density, that the free boundary is porous with porosity \(\delta >0\) and has finite \((N-\delta )\)-Hausdorff measure. Under a geometric compatibility condition set up by Rossi and Teixeira, it is established the behavior of a \(\ell \)-quasilinear optimal design problem with volume constraint for \(\ell \) small. As \(\ell \rightarrow 0^+\), we obtain a limiting free boundary problem driven by the infinity-Laplacian operator and find the optimal shape for the limiting problem. The proof is based on a penalization technique and a truncated minimization problem in terms of the Taylor polynomial of \(\varPhi \).

对于\（\ varPhi \）不满足Lieberman引入的自然条件的情况，考虑了在Orlicz–Sobolev空间中涉及\（\ varPhi \） - Laplacian的具有体积约束的优化问题。甲极小\（U_ \ varPhi \）在自由边界具有非简并证明存在和一些重要的后果建立等的李氏规则\（U_ \ varPhi \）沿着自由边界，即所述组\（\ {u_ \ varPhi> 0 \} \）具有均匀的正密度，自由边界是多孔的，孔隙率为\（\ delta> 0 \），并且具有有限的\（（N- \ delta）\）-豪斯多夫测度。下通过罗西和特谢拉建立一个几何相容条件，可以建立一个的行为\（\ ELL \） -quasilinear优化设计问题与卷约束\（\ ELL \）小。作为\（\ ell \ rightarrow 0 ^ + \），我们获得了由无穷拉普拉斯算子驱动的极限自由边界问题，并找到了该极限问题的最佳形状。该证明基于惩罚技术和根据\（\ varPhi \）的泰勒多项式的截断最小化问题。