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My text says:

In a paper in the Journal de Mathématiques for 1890, Picard considered the linear second-order partial differential equation $$A=0,$$ in which the coefficients are functions of $x$ and $y$ in a domain for which $B^2 - AC < 0$. He showed that the solutions are determined by their values on the boundary of the domain, provided the domain is suitably small (a condition that ensures that the solution is single-valued and is therefore a function of $x$ and $y$).

He took the equation in the form $$\Delta u=u_{xx}+u_{yy}=F(u, u_x, u_y, x, y).$$

I have no idea what it's trying to say. It seems to me that $A$ itself is a second-order linear PDE, say $$A(z) = Rz_{xx} + Sz_{xy} + Tz_{yy} + Pz_x + Qz_y + Zz = 0,$$ where $R, S, T, P, Q, Z$ are functions of $x$ and $y$. But then it mentions some domain satisfying $B^2-AC<1$, and I have no idea what $B$ and $C$ are supposed to be for the inequality to make sense.

In the last line, are we taking $A(u) = u_{xx}+u_{yy}$?

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    Where did you take this from? Maybe the context will help us, because, indeed, this looks extremely confusing. – 2017-01-12
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    It's from the lecture notes for a course under a section titled "Picard on second-order linear elliptic equations". – 2017-01-12

2 Answers 2

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Even if you don't speak French, just looking at the formulae on the first two pages of Picard's memoir should be enough to convince yourself that the equation under discussion is

$$A \frac {\partial^2 u} {\partial x^2} + 2B \frac {\partial^2 u} {\partial x \partial y} + C \frac {\partial^2 u} {\partial y^2} = F \left( u, \frac {\partial u} {\partial x}, \frac {\partial u} {\partial x}, x, y \right)$$

with exactly the conditions given to you: $A, B, C$ functions of $x$ and $y$ with $B^2 - AC$ having constant sign on the domain under consideration.

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    Thanks for the link to the original paper source. How did you find it? – 2017-01-12
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    @IrregularUser: If I tell you, you'll feel embarassed, so I'll just leave [a link](https://www.google.com/search?q=Journal+de+Mathématiques+1890+Picard) - for me, it's the first result in the list... Notice that the French have digitized all of their historically important mathematics works, which is quite a feat! They really show that you don't find only cat pics on the internet. – 2017-01-12
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    Wow. I searched the exact same thing and must have not clicked that - every other link on the first page is clicked! That's actually very impressive (a tool that I'll use to check sources in the future), and I hear that they have streets/roads named after mathematicians too. – 2017-01-12
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    @IrregularUser: *"I hear that they have streets/roads named after mathematicians too"* - comments are not for chatting but I can't help it: why are you so surprised? Doesn't this happen in all countries with some cultural history (naming streets and squares after scientists and artists)? It surely does in my country. – 2017-01-12
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    I guess I shouldn't be surprised. A lot of maths took place in Paris and Berlin, so it perhaps it's natural that places get named after mathematicians at the time. Then again, I'm not geographically oriented and don't really know places outside of my own city. – 2017-01-12
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Apparently the text has a rather awful typo, almost completely obliterating the PDE. My guess is that the PDE is supposed to be $A u_{xx} + 2 B u_{xy} + C u_{yy} = 0$, where $A,\; B,\; C$ are functions of $x$ and $y$.