# References on the obstacle problem for the heat equation

Can you point out some references that deal with the obstacle problem for the heat equation?

$$(OP) \quad\begin{cases} \max\{\Delta u -\partial_t u, \varphi - u \} = 0 & \text{ in } (0,T)\times \mathbb{R}^n \\ u(0,\cdot) = \varphi(0,\cdot) & \text{ in } \mathbb{R}^n \end{cases}.$$

Works on elliptic obstacle problems appear to be much easier to find (see Wikipedia, for instance).

Since a bounty has been offered for this question, I'll write down what I feel is missing in the current (very nice) answer and that I'd like to see:

• complete argument for the existence (with references too)
• further details on the representation of solutions using the heat kernel
• references about numerical analysis of the problem (and Matlab/Mathematica codes)
• references on physical motivations for the problem.
• @parsiad Techniques to prove uniqueness of strong solutions; if the unique strong solutions can be written using the heat kernel. – user108310 Jul 29 '17 at 10:04

I will get you started, but there are lots of blanks to fill. We are interested in the PDE \begin{align*} \min\left\{ -u_{t}+\Delta u,u-\varphi\right\} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\ u(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n} \end{align*} where $\varphi$ is a smooth function of polynomial growth.

## Existence

Let $u^{0}$ be a classical solution of \begin{align*} -u_{t}^{0}+\Delta u^{0} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\ u^{0}(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n} \end{align*} that is unique in an appropriately picked space of functions of polynomial growth. Define inductively $u^{k}$ as a classical solution of \begin{align*} -u_{t}^{k}+\Delta u^{k}+k\min\left\{ u^{k-1}-\varphi,0\right\} & =0 & \text{in }(0,T]\times\mathbb{R}^{n}\\ u^{k}(0,\cdot)-\varphi(0,\cdot) & =0 & \text{in }\mathbb{R}^{n} \end{align*} that is unique in an appropriately picked space of functions of polynomial growth. Now you would have to show that you can pass to limits ($k\rightarrow\infty$) to obtain a solution of the original PDE. The limiting solution is not, in general, twice differentiable in space. But, you should probably be able to establish sufficient conditions for it to be a once differentiable viscosity solution. This technique is called a penalty method and is at least useful for establishing existence.

## Uniqueness

Uniqueness can be handled in the space of viscosity solutions using a comparison principle argument. I wrote about this in an expository post, but I'm sure this is available in different forms elsewhere. The setting of the post proves uniqueness in the space of bounded functions, but you should be able to generalize the arguments.

## Heat Kernel

You can use smooth pasting to write the solution with the heat kernel, but you will probably not be able to find a "nice expression" for the free boundary, as this is believed to be hard.

## References

Van Moerbeke, Pierre. "On optimal stopping and free boundary problems." Archive for Rational Mechanics and Analysis 60.2 (1976): 101-148.

Pham, Huyên. "Optimal stopping, free boundary, and American option in a jump-diffusion model." Applied mathematics & optimization 35.2 (1997): 145-164.

• This is a useful answer. Thank you. If you find some other references or information, don't hesitate to add them. – user108310 Jul 29 '17 at 20:14
• Also, I've asked a related question (mathematica.stackexchange.com/questions/152491/…) to which you may be able to contribute. – user108310 Jul 29 '17 at 20:15
• Do you have a reference for the statement "You can use smooth pasting to write the solution with the heat kernel"? – user108310 Jul 30 '17 at 12:40
• For smooth pasting, see: Seydel, Rüdiger, and Rudiger Seydel. Tools for computational finance. Vol. 3. Berlin: Springer, 2006. The author deals with American options therein, but the technique works more broadly on appropriate $\varphi$. – parsiad Jul 30 '17 at 14:03
• Thanks a lot. I'll check that book. Also, since a bounty has been offered for the question, I updated it writing down what I feel is missing and would like to see in addition. – user108310 Aug 2 '17 at 8:37