论文标题
$γ$ - 具有可变指数的幂律功能
$Γ$-convergence for power-law functionals with variable exponents
论文作者
论文摘要
我们研究功能的$γ$ - convergence $ f_n（u）：= || f（\ cdot，u（\ cdot），du（\ cdot））|| _ {p_n（\ cdot）} $和$ \ Mathcal {f} _n（u）：= \int_Ω\ frac {1} {1} $ x \ in \ {l^1（ω，\ mathbb {r}^d），l^\ infty（ω，\ mathbb {r}^d），c（ω，\ mathbb {r}^d） $ w^{1，p_n（\ cdot）}（ω，\ mathbb {r}^d）$。这里$ω\ subseteq \ mathbb {r}^n $是一个有限的开放集，$ n，d \ ge 1 $和可测量的函数$ p_n：\overlineΩ\ rightarrow（1， + \ frty）（ + \ iffty）$满足条件$ \，β\，{\ Mathop {\ rm ess \：inf}}} _ {\ \ \ \ edlineω} p_n $对于固定常数$β> 1 $和$ {\ MATHOP {\ MATHOP {\ rm ess \：inf}}}}}} _ { \ infty $。我们表明，当$ f（x，u，\ cdot）$是级别凸且较低的半连续性时，它可以从下面满足均匀的生长条件，然后，作为$ n \ to \ infty $，序列$（f_n）_n $ $ $ $ $ $ $ $ $ $ $umγ$ -converges in $ x $ in $ x $ to $ x $ to $ f $ f $表示为$ f（$ f（$ f（$ f（u）$ f） f（\ cdot，u（\ cdot），du（\ cdot））|| _ {\ infty} $上的有效域$ w^{1，\ infty}（ω，\ mathbb {r}^d）$。此外，我们表明$γ$ - $ \ lim_n \ mathcal f_n $由功能$ \ Mathcal {f}（u）给出：= \ left \ webt \ {\ stray {array} {lll} {lll} {lll} \ \！\！\！\！\！\！\！\！\！\！\！\！ ＆0＆\ hbox {if} || f（\ cdot，u（\ cdot），du（\ cdot））|| _ {\ infty} \ leq 1，\\ \ \！\！\！\！\！\！\！ ＆ +\ infty＆\ hbox {否则在} x。\\ \ end {array} \ right。 $
We study the $Γ$-convergence of the functionals $F_n(u):= || f(\cdot,u(\cdot),Du(\cdot))||_{p_n(\cdot)}$ and $\mathcal{F}_n(u):= \int_Ω \frac{1}{p_n(x)} f^{p_n(x)}(x,u(x),Du(x))dx$ defined on $X\in \{L^1(Ω,\mathbb{R}^d), L^\infty(Ω,\mathbb{R}^d), C(Ω,\mathbb{R}^d)\}$ (endowed with their usual norms) with effective domain the Sobolev space $W^{1,p_n(\cdot)}(Ω, \mathbb{R}^d )$. Here $Ω\subseteq \mathbb{R}^N$ is a bounded open set, $N,d \ge 1$ and the measurable functions $p_n: \overlineΩ \rightarrow (1, + \infty) $ satisfy the conditions ${\mathop{\rm ess\: sup }}_{\ \overline Ω} p_n \le \, β\, {\mathop{\rm ess\: inf }}_{\ \overline Ω} p_n $ for a fixed constant $β> 1$ and $ {\mathop{\rm ess\: inf }}_{\ \overline Ω} p_n \rightarrow + \infty$ as $n \rightarrow + \infty$. We show that when $f(x,u,\cdot)$ is level convex and lower semicontinuous and it satisfies a uniform growth condition from below, then, as $n\to \infty$, the sequences $(F_n)_n$ $Γ$-converges in $X$ to the functional $F$ represented as $F(u)= || f(\cdot,u(\cdot),Du(\cdot))||_{\infty}$ on the effective domain $W^{1,\infty}(Ω, \mathbb{R}^d )$. Moreover we show that the $Γ$-$\lim_n \mathcal F_n$ is given by the functional $ \mathcal{F}(u):=\left\{\begin {array}{lll} \!\!\!\!\!\! & 0 & \hbox{if } || f(\cdot,u(\cdot),Du(\cdot)) ||_{\infty}\leq 1,\\ \!\!\!\!\!\! & +\infty & \hbox{otherwise in } X.\\ \end{array}\right. $
