Boundedness and higher integrability of minimizers to a class of two-phase free boundary problems under non-standard growth conditions

Abstract

In this paper, we are concerned with the existence, boundedness, and integrability of minimizers of heterogeneous, two-phase free boundary problems $ \mathcal {J}_{\gamma}(u) = \int_{\Omega}\left(f(x, \nabla u)+\lambda_{+}(u^{+})^{\gamma}+\lambda_{-}(u^{-})^{\gamma}+gu\right)\text{d}x \rightarrow \text{min} $ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $ \gamma = 0 $, chemical reaction problems with $ 0 < \gamma < 1 $, and obstacle type problems with $ \gamma = 1 $, respectively.

Keywords

• free boundary problem
• two-phase
• integrability
• non-standard growth
• minimizer