We consider an optimization problem with volume constraint for an energy functional associated to an inhomogeneous operator with nonstandard growth. By studying an auxiliary penalized problem, we prove existence and regularity of solution to the original problem: every optimal configuration is a solution to a one phase free boundary problem—for an operator with nonstandard growth and non-zero right hand side—and the free boundary is a smooth surface.

中文翻译：

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具有非标准增长的非均质算子的具有体积约束的优化问题

我们考虑与具有非标准增长的非均匀算子相关的能量函数的体积约束的优化问题。通过研究辅助惩罚问题，我们证明了原始问题的解决方案的存在性和规律性：对于具有非标准增长且右手非零的算子，每个最优配置都是一个单相自由边界问题的解决方案。边界是光滑的表面。