Question

Example of fundamental principle of malgrange palamodov

Answer 1

Answer:

In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956).

This means that the differential equation

{\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\cdots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=\delta (\mathbf {x} ),} P\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_{\ell}}\right)u(\mathbf{x})=\delta(\mathbf{x}),

where P is a polynomial in several variables and δ is the Dirac delta function, has a distributional solution u. It can be used to show that

{\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\cdots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=f(\mathbf {x} )} P\left(\frac{\partial}{\partial x_1}, \cdots, \frac{\partial}{\partial x_{\ell}}\right)u(\mathbf{x})=f(\mathbf{x})

has a solution for any compactly supported distribution f. The solution is not unique in general.