Solvability for constant-coefficient partial differential operators - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T05:32:05Z http://mathoverflow.net/feeds/question/118546 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118546/solvability-for-constant-coefficient-partial-differential-operators Solvability for constant-coefficient partial differential operators Kevin McLeod 2013-01-10T16:06:08Z 2013-01-11T08:27:08Z <p>Let $\mathcal{S}$ denote the space of Schwartz functions on $\mathbb{R}^n$, and $\mathcal{S}'$ the space of tempered distributions. Let $L$ denote a linear, constant-coefficient, partial differential operator. I would like to know if there is a "simple" proof of solvability for $L$; $\textit{i.e.}$ of the result that for any $f \in \mathcal{S}'$ there is some $u \in \mathcal{S}'$ such that $Lu = f$. I am interested in the result for general $L$, but would be satisfied with a proof that worked for the classical Laplace, wave and heat operators.</p> <p>Here is what I know so far:</p> <p>1) If $L$ has a fundamental solution, $G \in \mathcal{S}'$ such that $LG = \delta$ (the Dirac delta), then $Lu = f$ is solvable for any $f$ such that the convolution $f \ast G$ exists (in $\mathcal{S}')$. This is not obviously equivalent to solvability for arbitrary $f$, though a "simple" proof of such equivalence, even for the classical operators, would answer my question.</p> <p>2) Hormander and Lojasiewicz answered the general problem of solvability affirmatively in 1958. (Any "simple" proof therefore has to be simpler than Hormander's.)</p> <p>3) Bernstein, in the 1970's, used analytic continuation to show the existence of a fundamental solution $G \in \mathcal{S}'$ for any $L$.</p> <p>4) There have subsequently been several "simple" proofs of the existence of fundamental solutions, though these are often located in the space $\mathcal{D}'$ of general distributions.</p> <p>A possible re-statement of my original question is therefore: can Bernstein's method of analytic continuation, or any of the other "simple' proofs of the existence of a fundamental solution, be extended to answer the general solvability problem in $\mathcal{S}'$?</p> http://mathoverflow.net/questions/118546/solvability-for-constant-coefficient-partial-differential-operators/118608#118608 Answer by Bazin for Solvability for constant-coefficient partial differential operators Bazin 2013-01-11T08:27:08Z 2013-01-11T08:27:08Z <p>For your constant coefficient operator $L(D)$, you want a fundamental solution $G$ such that $\hat G$ is a multiplier of $\mathcal S'$. This is not even true for the Laplace equation: the fundamental solution is, in more than 3D, $a_d\vert x\vert^{2-d}$ and its Fourier transform is $$b_d\vert \xi\vert^{-2}=\hat G(\xi).$$ The map $\mathcal S\ni\phi\mapsto \phi \hat G\in L^1$ is well defined but $\hat G$ is not a multiplier of $\mathcal S$ or $\mathcal S'$.</p> <p>On the other hand, if you consider $L(D)=-\Delta +m$ with $m>0$, you find $$\hat G=c_d(m+\vert \xi\vert^2)^{-1}$$ which is indeed a multiplier of $\mathcal S$ and for this equation you have your solvability property.</p>