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I have the following equation involving the derivative of a 2-dimensional Dirac delta function

$$\partial_x[\alpha(x,y)\delta(x^2 f(x,y)+g(x,y)+h(y)^2)]=0,$$

where $\alpha$ is a unknown function coming from solving an homogenous equation, $f$ and $g$ are distributions, and $h$ is a scalar function.

Expanding out would suggest $$\delta(x^2 f(x,y)+g(x,y)+h(y)^2)\,\partial_x \alpha(x,y)+\alpha(x,y)\,\partial_x\delta(x^2 f(x,y)+g(x,y)+h(y)^2)=0.$$

Can I here integrate and use the identity $$\int\delta'(x)\phi(x)dx=-\int\delta(x)\phi'(x)dx,$$

which obviously would satisfy the equation. Or does this not hold in my case of a multivariable delta function?

We can also expand the derivative on the delta function using the chain rule, but I don't know if that helps.

I am trying to find some constraints on $\alpha$, so it would be ideal if it didn't satisfy it in this way and I could instead extract some information about $\alpha$.

Bernard
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gammadragon
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1 Answers1

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Without additional regularity on $f$ and $g$ the equality is ill defined in the sense of distributions. See this question for more on this. Let's compute the second term in your second equation. Take a test function $\phi \in \mathcal{C}_c^\infty (\mathbb{R}^2)$ and define $k(x,y) = x^2 f(x,y) + g(x,y) +h(y)^2$. Then, allowing an abuse of notation we can write:

$$ \begin{aligned} \langle \alpha \partial_X (\delta \circ k), \phi \rangle &= \int \partial_X (\delta \circ k) \phi \alpha \; dxdy \\ & = -\int \delta \circ k \; \partial_x (\phi \alpha) \; dx dy \\ & = -\int \delta \; [\partial_x (\phi \alpha) \circ k^{-1}] |\text{det } D k^{-1}| \; dx dy \end{aligned} $$ where $\delta \circ k$ denotes the composition of the Dirac delta with the function $k$. Note that we require $k$ and its inverse to be smooth for the last equality to hold. As for $\alpha$, it needs to be smooth as well for the problem to be well-defined. The first term in your second equation can be computed similarly.

Saleh
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