Libertarianism - the variants I am familiar with - holds to what it pleases its advocates to call "the non-aggression axiom." This axiom may be stated as "thou shalt not initiate the use of force against non-aggressors." It may be restated as "thou mayest initiate the use of force only against aggressors."
Libertarianism - again, the variants I am familiar with - also holds to the concept of private property. Let us define private property as an agreement between agents that some shit is said to belong to one agent, and other shit might belong to another agent. One person might be said to own all the shit, as well - everyone can agree that everything belongs to one person. The narratives which justify the division of the shit may vary, but in the end, people are calling some shit "mine". And aggression, in the variants of Libertarianism that I am familiar with, includes the concept of aggression against property - or, in other words, "touching my shit" counts as aggression.
Now, watch this magic. The non-aggression axiom combined with private property - that is to say, Libertarianism - may be reduced to "if you touch what I call my shit, I get to kill you."
But that's how private property is already approached in damn near every non-Libertarian governmental structure on the planet Earth: if you touch my shit, I get to kill you; not only that, I can call the forces of society to join me against you, and you can't call them against me. And "touching my shit" is the primary justification for the application of force in human relationships already - aggressing against aggressors. Therefore, Libertarianism solves nothing, but does have the benefit of making the root of our problem - private property - more obvious than it would otherwise be.
And, here's more fun. If the libertarian is an atheist, he's in the position of claiming we arrived at our current totalitarian state from a stateless - libertarian - society in the distant past, where shit and claims thereto were the sole motivators of force between humans - in other words, people were killing each other over claims to shit, with individual conflict resolution over shit evolving into group conflict resolution over shit evolving into the state. So, on the evolutionary paradigm, libertarianism obviously negates itself, and is therefore false.
And if the libertarian is a Christian, then he's got to explain the Law of Moses - it's not obvious, to put it mildly, that God's a libertarian. And they also must explain the multiple injunctions to submit to one's rulers throughout the New Testament, beginning with Christ's teaching to render unto Caesar the things which are Caesar's, with specific reference to money issued by the state.
Libertarianism seems to me to be simply a narrative of justification for covetousness - the desire to possess shit - and other anti-social behaviors. The problem they're solving isn't how to have peace among men, but how to justify reapportioning the shit more to their liking. The philosophy negates itself, giving rise to its antithesis, the state, therefore it is false.
But, by the same token, the state negates itself, too - therefore it, too, is false.
The root problem is the necessary negation of liberty entailed by private property.