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I have a question on the introduction to Hormanders first PDE book. The introduction seems poorly (i.e. confusingly) written to me, hopefully the rest of the book is better. Anyway, he says classical solutions of the wave equation $$ \frac{\partial^2}{\partial x^2}v - \frac{\partial^2}{\partial y^2}v = 0, $$ are twice continuously differentiable functions satisfying the equation everywhere. These solutions are functions of the form $$ v(x,y) = f(x+y) + g(x-y), \quad \quad (*) $$ where $f$ and $g$ are twice continuously differentiable. So far so good. He then says that classical solutions have as uniform limits all functions of the form $(*)$ with $f$ and $g$ continuous. All such functions ought to be recognized as solutions of the wave equation so therefore the definition of a classical solution is too restrictive.

So is Hormander saying that if we take a sequence of classical solutions $\{v_n\}$ such that $v_n(x,y) = f_n(x+y) + g_n(x-y)$ with $f_n$ and $g_n$ twice continuously differentiable, then the uniform limit of the sequence $\{v_n\}$ may be just continuous? He says 'uniform limit' but uniform with respect to what, is he about talking pointwise uniform convergence? Finally he seems to be saying that we should recognise certain merely continuous functions as solutions of the wave equation..surely this can't be correct, we need functions to be at least continuously differentiable in order to have weak solutions?

Can anybody clarify this section in Hormanders book for me?

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    $\begingroup$ Generalized functions which are solutions of the PDE need not be differentiable. They need not even be continuous. They can be just locally integrable, for example. $\endgroup$ – Kavi Rama Murthy Aug 1 '18 at 6:42
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    $\begingroup$ Uniform means just ordinary uniform convergence. What on earth do you mean by “pointwise uniform convergence”? $\endgroup$ – Hans Lundmark Aug 1 '18 at 8:23

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