关于分层李群上具有奇异性的广义 Prandtl-Batchelor 自由边界问题

On a class of elliptic free boundary problems with multiple solutions

We prove that a certain class of elliptic free boundary problems, which includes the Prandtl-Batchelor problem from fluid dynamics as a special case, has two distinct nontrivial solutions for large values of a parameter. The first solution is a global minimizer of the energy. The energy functional is nondifferentiable, so standard variational arguments cannot be used directly to obtain a second nontrivial solution. We obtain our second solution as the limit of mountain pass points of a sequence of $C^1$-functionals approximating the energy. We use careful estimates of the corresponding energy …

Regularity in the two-phase free boundary problems under non-standard growth conditions

In this paper, we prove several regularity results for the heterogeneous, two-phase free boundary problems $\mathcal {J}_γ(u)=\int_Ω\big(f(x,\nabla u)+λ_{+} (u^{+})^γ+λ_{-}(u^{-})^γ+gu\big)\text{d}x\rightarrow \text{min}$ under non-standard growth conditions. Included in such problems are heterogeneous jets and cavities of Prandtl-Batchelor type with $γ=0$, chemical reaction problems with $0<γ<1$, and obstacle type problems with $γ=1$. Our results hold not only in the degenerate case of $p> 2$ for $p-$Laplace equations, but also in the singular case of $1<p<2$, which are extensions of \cite{Ld…

Feynman-Lagerstrom 边界层准则

The Feynman-Lagerstrom criterion for boundary layers

本文研究微粘性定常流动的边界层理论，证明 Feynman-Lagerstrom 涡量选择条件是周期 Prandtl 边界层存在的充分条件，并通过精细迭代与高阶能量方法处理隐式选择机制。

Regularity for degenerate two-phase free boundary problems

We provide a rather complete description of the sharp regularity theory for a family of heterogeneous, two-phase variational free boundary problems, $\mathcal{J}_γ\to $ min, ruled by nonlinear, $p$-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl-Batchelor type; singular degenerate elliptic equations; and obstacle type systems. The Euler-Lagrange equation associated to $\mathcal{J}_γ$ becomes singular along the free interface $\{u= 0\}$. The degree of singularity is, in turn, dimed by the parameter $γ\in [0,1]$. For $0< γ< 1$ we show local…