Can the coefficient of static friction be less than that of kinetic friction? I was recently wondering what would happen if the force sliding two surfaces against each other were somehow weaker than kinetic friction but stronger than static friction. Since the sliding force is greater than the maximum force of static friction, $F > f_s = \mu_s F_N$, it seems that the surfaces should slide. But on the other hand, if the force of kinetic friction is greater than the applied force, there'll be a net force $\mu_k F_N - F$ acting against the motion, suggesting that the surfaces should move opposite to the direction they're being pushed! That doesn't make sense.
The only logical resolution I can think of is that the coefficient of static friction can never be less than the coefficient of kinetic friction. Am I missing something?
 A: This answer is speculative - not based on my experience with friction.
Logically, there's no reason kinetic friction has to be velocity-independent.  You could have kinetic friction that increases with velocity.  That way, if you push on something with more force than static friction, the thing would accelerate up to some certain velocity at which kinetic friction equaled the applied force, and then accelerate no more.
If that speed were very slow, you could say that kinetic friction is greater than static friction for all normally-encountered speeds without a paradox.
However, as you pointed out, kinetic friction would have to be less than or equal to static friction for speeds right next to zero.
A: The problem with this question is that static friction and kinetic friction are not fundamental forces in any way-- they're purely phenomenological names used to explain observed behavior. "Static friction" is a term we use to describe the observed fact that it usually takes more force to set an object into motion than it takes to keep it moving once you've got it started.
So, with that in mind, ask yourself how you could measure the relative sizes of static and kinetic friction. If the coefficient of static friction is greater than the coefficient of kinetic friction, this is an easy thing to do: once you overcome the static friction, the frictional force drops. So, you pull on an object with a force sensor, and measure the maximum force required before it gets moving, then once it's in motion, the frictional force decreases, and you measure how much force you need to apply to maintain a constant velocity.
What would it mean to have kinetic friction be greater than static friction? Well, it would mean that the force required to keep an object in motion would be greater than the force required to start it in motion. Which would require the force to go up at the instant the object started moving. But that doesn't make any sense, experimentally-- what you would see in that case is just that the force would increase up to the level required to keep the object in motion, as if the coefficients of static and kinetic friction were exactly equal.
So, common sense tells us that the coefficient of static friction can never be less than the coefficient of kinetic friction. Having greater kinetic than static friction just doesn't make any sense in terms of the phenomena being described.
(As an aside, the static/kinetic coefficient model is actually pretty lousy. It works as a way to set up problems forcing students to deal with the vector nature of forces, and allows some simple qualitative explanations of observed phenomena, but if you have ever tried to devise a lab doing quantitative measurements of friction, it's a mess.)
A: With the simple model of friction, using coefficients of static and kinetic friction, you are right - the coefficient of kinetic friction cannot be higher than the coefficient of static friction.
In real world, the phenomenon of friction may be much more complex. Your reasoning points out that with the velocity very close to zero kinetic friction cannot be much higher than static friction. To be more accurate, if the velocity is infinitely small, the kinetic friction may be higher than static one, but then it may only be infinitely small! An example is air friction, which is zero when a body doesn't move and increases with velocity. In a simple model, for small velocities, air friction is just proportional to velocity.
A: The kinetic friction is usually not greater than the applied force. 
You start moving an object which increases the static force to prevent any motion. However, you keep applying more force until you reach a maximum value for the static friction. Then, the object begins to move. However, the kinetic friction (which is produced when the object moves) is less than the applied force. Furthermore, the kinetic friction doesn't depend on the applied force, therefore, you don't need to apply more force to keep moving the object (actually, you need less force).
A: With static friction there is no movement between the surfaces and therefore no work is done, and so the two surfaces remain at the same temperature (ambient).
However, with dynamic friction there is work done and so one or both surfaces absorb the energy and heat up. With typical friction materials the maximum coefficient of friction is only achieved when the material is heated, and in many (not all) cases the dynamic coefficient of friction, at elevated temperature and with suitable contact pressure and sliding speed, is in fact HIGHER than the static coefficient of friction with lower (ambient) temperature and zero sliding speed with the same contact pressure. 
Certainly for some types of materials it is demonstrably the case that the static friction coefficient can be lower than the quoted dynamic friction coefficient. 
A: I think the answer is yes. Here's one possible explanation for it that's only speculation. According to https://physics.aps.org/story/v7/st6, the coefficient of static friction between mismatched smooth surfaces is zero. Surely that means the coefficient of static friction can be less than that of kinetic friction for a given normal force. How can that be? I think the theory predicts that for sufficiently low sliding speeds of smooth surfaces, the force of kinetic friction per area varies linearly with sliding speed. In order to prove that the coefficient of static friction can't be less than that of kinetic friction, you have to assume that the force of kinetic friction per area is independent of the sliding speed.
