Elsa & Anna, the Norwegian sisters
Beauty
Based on corpus linguistics, it can be seen that beauty is very rarely used to refer to the two main females. It more often refers to ice and snow.
The first time Anna is called beautiful, it is to set up a joke, which shows her decidedly not beautiful self.
Disney's first realistic looking princess
The second time either of them is called beautiful, it is to each other while trying to engage in small talk. In this scenario, neither of them is objectified as something to be admired by the male gaze.
Even if they are so unrealistically beautiful that they have heads wider than their hips, they are not mere objects.
The closest the film comes to objectifying them is by changing Elsa's outfit to a slinky dress with a slit showing off her legs. However, the scene where she gets the dress and her sexiest pose with it are in a moment of character growth, which when seen through the lens of third-wave feminism, is symbolic of her growing and taking control over her sexuality. In addition, after this scene she never struts any more poses for the audience to ogle, supporting the idea that the dress was a symbol of her character growth.
And also, is this really that sexy to anyone?
Passivity
In terms of passivity, both characters are active participants in their own stories. This is most evident in the corpus,
where the words “I’ll” and “We’ll” show up fairly often, and often said by the two protagonists. This indicates that they are at least making an effort to get what they want, instead of waiting for their prince to get it for them.
When Elsa’s powers are revealed, she flees to the mountains to protect her kingdom and Anna immediately goes after her, both actively pursuing their goals.
It's only been the 1st act, and these two have shown more balls than Snow White, Cinderella and Aurora combined.
Further, the motivation for both of them is each other, not any male character, further severing them from the control of the patriarchy.
A love so strong, it caused Elsanna to be the top lesbian pairing on Fanfiction.net
Female Voice in the Patriarchy
And about that patriarchy, there isn’t really one. Elsa and Anna are born into power as royalty and Elsa in particular is groomed to become the future ruler of their kingdom. Neither of them is forced into stereotypically feminine roles and both are not hindered by any sort of white male system.
Your patriarchy has no power in our realm, you puny little man
If anything, Elsa’s emotional detachment is a sign of her severing her connection to the feminine voice as she becomes more like a Byronic hero. By learning to accept her emotions, she reconnects to the feminine voice.
Then: Emotions are for girly girls anyway
Now: Having emotions is awesome! I love being a girl!
Anna has no issue accessing the power granted by the patriarchy or retaining her feminine voice. Her story incoporates both her femininely caring heart and her masculine proactivity, and her character growth does not involve gender roles in any way.
I'm gonna be my awesome girly self and all you haters can just step back
Female Archetypes
In Frozen, the two main females Elsa and Anna at first seem to fit into the roles of the victim and villainess. However, in this story their characters are more developed and nuanced, allowing them to subvert and transcend the simple roles of damsel in distress and femme fatale.
Concept art of Elsa and Anna that shows them as more obvious archetypes and features a FABULOUS mink coat that better return in the sequel
Elsa holds great power, so as per the male gaze theory, she should be punished so the male viewer is not threatened by her power.
"She's an X-Man!"
"Quick, kill her before they make another crappy movie!"
However, she is not a simple villain who is punished, but rather a sympathetic victim as well, forced into her postion of antagonist by circumstance rather than choice.
Spurned by my kingdom, hunted by my enemies and abandoned by my family ...
... well, might as well break out the Broadway showtune
Through the event of the film, she learns not to bury her emotions and become more caring, becoming the transvestite heroine, possessing masculine power and feminine emotions.
True love will thaw a frozen heart <3
Anna’s character takes the opposite route, starting out as a naïve, very feminine girl who, based on patterns of previous works, is likely to be the victim that would need to be saved by the male hero.
Spoiler: This relationship does not end well
She does end up needing to be saved but instead sacrifices herself to save Elsa.
Told you!
She too grows up and becomes the transvestite heroine, subverting the expectations of the audience.
Now I'm a girly girl who can still kick ass