I recently learned about Dirichlet problems and was wondering if there were similar solutions in the case where only few temperature points are known instead of a continuous temperature boundary.


For instance, say the temperature is known at the three points of an equilateral triangle, and is assumed to be at a steady state. Is it possible to derive a differential (and of closed form) equation that describes the temperature within the triangle?


Additionally, what are these type of problems called? 



Point boundary problems
I think you may have gotten things backwards. 


The point of differential equations is to describe macroscopic (global) phenomena via microscopic (local) physical laws, as differential operators are strictly local objects. Solving a differential equation one often finds families of solutions, which can live in various different function spaces of different regularity. The study of well-posedness of a differential equation often becomes the study of *under what conditions can we obtain the existence of a unique solution*. It is often found that the "degrees of freedoms" in the families of solutions can be restricted by prescribing boundary data of sufficient regularity. This is the case of elliptic operators and leads to the Dirichlet problem. 


Now, the amount of data to be prescribed at the "boundary" is not the same for every problem. For elliptic type problems one only need to give a Dirichlet or Neumann type condition, but in general giving both conditions may lead to non-existence of a solution (over-determined problem). But for a different type of boundary and a different type of equation (say hyperbolic/wave equation in the initial value problem formulation), it is necessary to prescribe both the "Dirichlet" and the "Neumann" conditions for the question to be well-posed. 


What you are asking is sort of an opposite problem: you are asking that given the value of a function on some subset of, say, the plane, what differential equations are well-posed for this data. This problem has too many solutions. Just to give a few


* As I mentioned in the comments, the equation $\partial^2f = 0$ (vanishing Hessian)
* Or, let $v,w$ be arbitrary unit vectors not parallel to the sides of the triangle, then you can set $v(f) = w(w(w(f))) = 0$. The solution is constant in the $v$ direction, and along integral lines of $w$ must be quadratic, which is determined by three constants.
* Or, let $v$ be the unit vector going from point 1 to point 2, and let $w$ be the unit vector that connects point 3 to the line formed by point 1 and point 2 perpendicularly. Let $a$ be a number that is not a multiple of the distance between points 1 and 2, and $b$ be a number that is not a multiple of the distance between point 3 and the line, then take $v(v(f)) = - a^2 f$ and $w(w(f)) = - b^2 f$. The general solution is a trigonometric function depending on one translation coordinate and one scaling parameter.


But if you ask that also the function represents the steady state of a solution to the classical heat equation (in other words a solution to the Laplace equation), then the answer is no: you can extend the temperature profile on the boundary of your triangle arbitrarily from the three fixed data points you gave. For every extension (say differentiable) there is a corresponding solution to the Laplace equation. In other words, there are many, many steady-states whose temperature are as given at those three points on the triangle. So the map from your data to admissible steady-state temperature distributions is necessarily non-unique. So unless you prescribe additional conditions to pick out which of the many possible solutions you want, it is in general impossible (by definition) to write down a well-posed differential equation doing what you want it to do. 


*Edit: I just saw your answer to Qiaochu's comment*


Yes, the devil is in the details on how you insert the constraint though. The Dirichlet problem is well posed. The three-point version isn't. By counting dimensions your constraint needs to be strong enough to mod-out a infinite dimensional set. For example, my first example of an equation $\partial^2 f = 0$, is one possibility. A solution to that equation most certainly solves the Laplace equation. It is equivalent to extending the data to be linear along the boundary of the triangle, and solving the Laplace equation. It also happens to be the one that also minimizes the $H^2$ norm among all solutions to the Laplace equation. Is it a meaningful one? I dunno, what do you think? But it certainly is optimal when considering one metric.


In any case, any conditions you can impose that leads to a unique solution most certainly will be equivalent to one that leads to a unique set of compatible boundaries. Then you can impose differential conditions (ask that $f$ solves a second order ODE along each segment of the boundary) or algebraic conditions. Unless you have a physical justification, optimality really is in the eye of the beholder. 


