Solving Poisson's equation subject to a non-linear constraint

Question

Problem

Let the function $f \in (L^4 \cap H^1)(\Omega)$ satisfy the following pair of equations: $$-\Delta f = 2 \nabla \theta \cdot (J[\theta] \nabla \theta)\quad \text{ on }\Omega,$$ $$\nabla f \cdot \nabla \theta - \frac{1}{2}f^2 = \frac{1}{4}|\nabla \theta|^4 + |J[\theta]|^2\quad \text{ on }\Omega.$$ In the above pair of equations the function $\theta \in C^\infty(\overline{\Omega})$ satisfies $$-\Delta \theta = 0 \text{ on }\Omega,$$ $$\theta = g \text{ on }\partial \Omega,$$ where $g \in C^\infty(\partial \Omega)$ is given.

Notation clarification

To clarify notation, $J[\theta]$ is the Jacobian matrix of second derivatives of $\theta$ and we understand $$\nabla \theta \cdot (J[\theta] \nabla \theta) := \sum_{i,j=1}^N \frac{\partial \theta}{\partial x_i}\frac{\partial^2 \theta}{\partial x_i \partial x_j}\frac{\partial \theta}{\partial x_j}$$ $$|J[\theta]|^2 := \sum_{i,j=1}^N \left(\frac{\partial^2 \theta}{\partial x_i \partial x_j}\right)^2.$$ Additionally, $\nabla$ denotes the grad operator and $\Delta$ denotes the Laplacian operator div grad.

Issue

I wish to understand the behaviour of the function $f$ analytically, and potentially find a way to solve this problem. I know that if $f$ had given Dirichlet boundary conditions then it would be uniquely defined but I am unsure about this type of problem. My questions are:

1. What is the name of my second equation (the first order non-linear problem)?
2. Can I say anything about the existence or uniqueness of the solutions $f$?
3. How can I analytically understand the behaviour of $f$?
4. Can I solve this problem analytically?

Answer 1

1. I seriously doubt this equation has a name. The best I could say is that it is a semilinear transport equation.

2. Your problem is phrased in a way that makes it seem that $\theta$ is a known harmonic function (with the given Dirichlet data). In this case, let $h:= 2 \nabla \theta \cdot (J[\theta]\nabla \theta)$; then $-\Delta f=h$ in $\Omega$. This is simply Poisson's equation - the existence and uniqueness theory for Poisson's equation is very well-known and can be found in pretty much any textbook on PDE. However, with the second equation for $f$ the problem is most likely overdetermined which means you can probably only find solution in certain domains $\Omega$.

3. Again you need to look up Poisson's equation. For example, since $\theta$ is smooth, $h$ is smooth, so the theory for Poisson's equations says that $f$ is smooth inside $\Omega$ (not necessarily up to the boundary).

4. No. If $\Omega$ is arbitrary then you can't even solve for $\theta$.