奇异权超二次泛函的$Γ$ -收敛性
$Γ$-convergence of some super quadratic functionals with singular weights
摘要：
研究了如下泛函（$p>2$）$$ F_ε(u):=ε^{p-2}\int_Ω|Du|^p d(x,\partial Ω)^{a}dx+\frac{1}{ε^{\frac{p-2}{p-1}}}\int_ΩW(u) d(x,\partial Ω)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrtε}\int_{\partialΩ}V(Tu)d\mathcal{H}^2, $$的$Γ$ -收敛性，其中$Ω$是$\mathbb{R}^3$的开有界集，$W$和$V$分别是在$α, β$和$α', β'$处消失的两个非负连续函数.在前面的泛函中，我们固定$a=2-p$，$u$是一个标量密度函数，$Tu$表示它在$\partialΩ$上的迹，$d(x,\partial Ω)$表示到边界$\partial\Om$的距离函数。我们表明，奇异的能量$F_ε$的极限导致一个耦合的问题，体相和表面相变。
英文摘要：
We study the $Γ$-convergence of the following functional ($p>2$) $$ F_ε(u):=ε^{p-2}\int_Ω|Du|^p d(x,\partial Ω)^{a}dx+\frac{1}{ε^{\frac{p-2}{p-1}}}\int_ΩW(u) d(x,\partial Ω)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrtε}\int_{\partialΩ}V(Tu)d\mathcal{H}^2, $$ where $Ω$ is an open bounded set of $\mathbb{R}^3$ and $W$ and $V$ are two non-negative continuous functions vanishing at $α, β$ and $α', β'$, respectively. In the previous functional, we fix $a=2-p$ and $u$ is a scalar density function, $Tu$ denotes its trace on $\partialΩ$, $d(x,\partial Ω)$ stands for the distance function to the boundary $\partial\Om$. We show that the singular limit of the energies $F_ε$ leads to a coupled problem of bulk and surface phase transitions.
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