Example from Lewy

The example of Lewy is an example of a partial differential equation without smooth solutions, even though all data of the equation are smooth.

For a long time it was believed that at least for linear partial differential equations, an existence and uniqueness theory analogous to the theory of ordinary differential equations could be built up. The Cauchy-Kovalevskaya theorem (1875) seemed to point the way: every correctly posed Cauchy problem with analytical data has an analytical solution. Since the beginning of the 20th century, it has been possible to solve many partial differential equations and experience has shown that the differentiability properties of the data in the equation lead to differentiability properties of the solutions that may be influenced by the degree of the equation. It was therefore natural to assume that a statement analogous to Cauchy-Kowalewskaja's theorem applies when one goes from analytical functions to smooth functions. Lewy's surprisingly simple example refutes this assumption and Hans Lewy himself writes:

It was therefore a matter of considerable surprise to this author, to discover that this inference is in general erroneous.

German: The discovery that this conclusion is generally wrong was therefore a very big surprise for the author.

Lewy's example is a linear, partial differential equation of first order for complex-valued functions in three indeterminates : ${\displaystyle x_{1},x_{2},y_{1}}$

${\displaystyle {\Big (}-{\frac {\partial }{\partial x_{1}}}-i{\frac {\partial }{\partial x_{2}}}+2i(x_{1}+ix_{2}){\frac {\partial }{\partial y_{1}}}{\Big )}u=F}$.

In a first step, Lewy showed that if the right hand side of the same with only by and dependent and once continuously differentiable function when it is in an environment of is a once continuously differentiable solution, then in its analytical needs. Lewy used this to find a smooth function using Banach space arguments in a non-constructive way such that the above equation has no solution in , the latter being the space of all functions on whose first partial derivatives exist and a Holder condition Sufficient for all pairs of points . In particular, the linear partial differential equation has no smooth solution with this as the right hand side. ${\displaystyle \psi '(y_{1})}$ ${\displaystyle y_{1}}$ ${\displaystyle (0,0,y_{0})}$ ${\displaystyle \psi }$ ${\displaystyle (0,0,y_{0})}$ ${\displaystyle F(x_{1},x_{2},y_{1})}$ ${\displaystyle H^{1}}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \textstyle \leq 1}$ ${\displaystyle F}$

This example is of first order and only the coefficient before the derivative is non-constant, but as a polynomial (even first degree) very simple. The example of Lewy therefore also shows that the Malgrange-Ehrenpreis theorem cannot be generalized in a simple manner. ${\displaystyle y_{1}}$

Individual evidence

1. ^ Thomas Sonar: 3000 Years of Analysis. From counting stone to computer , Springer-Verlag 2011, chapter 10.8
2. ↑ Sophie Kowalevski: "On the theory of partial differential equation", Journal for pure and applied mathematics, Volume 80 (1875), pages 1–32
3. ↑ Hans Lewy (1957), An example of a smooth linear partial differential equation without solution , The Annals of Mathematics, Vol. 66, No. 1: pages 155–158
4. ^ GB Folland: Introduction to Partial Differential Equation , Princeton University Press 1995, ISBN 0-691-04361-2 , Chapter 1, Local Solvability, The Lewy Example