fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix Answer

plane geometry - Does the intersection of two moving curves sweep out a continuous line?

Hello dear visitor to our network We will offer you a solution to this question fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix ,and the answer will be typical through documented information sources, We welcome you and offer you new questions and answers, Many visitor are wondering about the answer to this question.

fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix

The Poisson equation is
$$\nabla^2 \Phi(\bar{r}) = 4\pi\rho(\bar{r})$$

The simplest potential is that of a point mass, for which the potential is of the form $-1/|\bar{r}|$ and the density function is the Dirac delta distribution $\delta(\bar{r})$. Now, my question is as follows: if I were to instead take the Hessian of that potential (eg in Cartesian coordinates), I should be able to take the trace and recover the Laplacian of the potential. So, I should find that the trace of the Hessian matrix gives me a delta distribution, but instead I find that the trace equals 0. I understand that this discrepancy arises from the fact that the Dirac distribution doesn’t behave like a normal function, but I’m curious how I would be able to resolve these seemingly inconsistent results. I’ve also tried computing the Hessian for this function in a basis-independent way using matrix calculus, but the results turn out to be the same.

Since $1/r$ is the fundamental solution of the Poisson equation, this is giving me quite a lot of trouble when I try to generalize computing the Hessian to more complicated potentials.

Sorry if this question isn’t well-phrased. My background is not in mathematics.

we will offer you the solution to fa.functional analysis – Poisson’s Equation, the Dirac Distribution, and the Hessian Matrix question via our network which brings all the answers from multiple reliable sources.