Before stating the question, I would like to first use an example for the type of formulation that I'm interested in.


Suppose we consider the continuity equation $\partial\_t \rho + \mathrm{div}( \rho v ) = 0$ with boundary conditions $\rho(0) = \rho\_0$ and $\rho(1) = \rho\_1$.
Now, if we were to test this equation with function $f$, we could arrive at the following equation (assuming zero boundary terms):
$$\int\_\Omega f ( \rho\_1 - \rho\_0 ) \ \mathrm{d}x = \int\_0^1 \int\_\Omega \nabla f \cdot (\rho v) \ \mathrm{d}x \ \mathrm{d}t.
$$


On the left side of the equation, we have values of $f$, while on the right, its gradient.  

By considering all $v$ and $\rho$ satisfying the continuity equation we could, in some sense, think that we are trying to "retrieve" $\nabla f$ back.




---


Question
--------


I would like to ask whether there are PDEs which could be used for "retrieving" the hessian $\nabla^2 f$ back in the above sense. (Or just in which hessian and not, for example, laplacian, appears explicitly)



Are there PDEs in which Hessian appears in the weak formulation
Integration by parts of $\nabla^{2} f$ against a symmetric field $\sigma=(\sigma\_{ij})$ yields eventually the formula,


$$
\int\_{\Omega}(\nabla^{2}f,\sigma)dx=\int\_{\Omega}f\,\nabla^{\*}\nabla^{\*}\sigma\,dx+\int\_{\partial\Omega}\partial\_nf\,(\sigma,n\otimes n)\,dS+\int\_{\partial\Omega}f\,T\sigma\,dS
$$
where $(\nabla^{\*}\sigma)\_{j}=\sum\_{i}\partial\_{i}\sigma\_{ij}$, $\nabla^{\*}\nabla^{\*}\sigma=\sum\_{i,j}\partial\_{i}\partial\_{j}\sigma\_{ij}$, and $T$ is a first order boundary operator mixing $\nabla^{\*}\sigma$ and the derivatives of $\sigma\cdot n$ on the boundary. Hence, requiring zero boundary conditions, you could cook up a "second order" continuity equation of the form $\partial\_{t}\rho-\nabla^{\*}\nabla^{\*}(\rho\sigma)=0$ in order to obtain,
$$
\int\_{0}^{1}\int\_{\Omega}(\nabla^{2}f,\rho\sigma)\,dx \,dt=\int\_{0}^{1}\int\_{\Omega}f\,\nabla^{\*}\nabla^{\*}(\rho\sigma)\,dx\,dt=\int\_{0}^{1}\int\_{\Omega}f\,\partial\_{t}\rho\,dx\,dt=\int\_{\Omega}f(\rho(1)-\rho(0))dt
$$
assuming that $f$ is either time independent or satisfies $f(0)=0$ and $f(1)=0$.


