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Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the Regularity Theorem for Elliptic Operators that if $f_{\alpha}$ are smooth and $f_{\alpha}$ are constants for $|\alpha|=N$, then $P(D)u$ is locally in $H_s(\Omega)$ if and only if $u$ is locally in $H_{s+N}(\Omega)$, where \begin{equation} H_s=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2\}. \end{equation}

I know almost nothing about partial differential equations, and am ignorant of special examples especially. I need some examples showing the conclusion is false for other types of differential operators. That is, something like $Lu=v$ where $v$ is good but $u$ is bad when $L$ is not elliptic.


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up vote 2 down vote accepted

Some trivial examples are always useful. We work on $\mathbb{R}^2$, with coordinates $(x,y)$. One can apply partial differential operators to distributions, and in particular to functions in $L^1_{\text{loc}}(\mathbb{R}^2)$.

  1. Take $D = \frac{\partial}{\partial x}$. Take any function as bad as you want, that depends only on $y$, and still belongs to $L^1_{\text{loc}}(\mathbb{R}^2)$. Then, $Du = 0$, the nicest, smoothest function you could possibly imagine, but still, $u$ is ugly.
  2. Take $D = \frac{\partial^2}{\partial x \partial y}$. Then any distribution $u$ has $Du = 0$, but this says nothing about $u$.
  3. Take $D = \frac{\partial^2}{\partial x^2} - \frac{\partial^2}{\partial y^2}$. The equation $Du = 0$ is the one dimensional wave equation. Any function $u$ of the form $u(x,y) = F(x - y) + G(x + y)$, where $F$ and $G$ are arbitrary, ugly as you want, functions solves $Du = 0$ formally. Not all such functions can be understood as distributions, but those that can, and for which you can perform integration by parts, yield examples of non-smooth distribution solutions. $D$ is an example of a hyperbolic operator.
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