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I'm trying to understand the notion of initial value problem and boundary value problem. I think I understand that, to have a well-posed problem we need to define a surface over a which we can define some data and from which we could obtain a unique solution, this would be a cauchy surface. My question is why for hyperbolic differential equation, the cauchy surface can be defined as a hypersurface at $t=0$ for example and for an elliptic differential equation, we need to define conditions on the boundary of our space, like for example the Laplace equation.

Stating it differently, why I could not define a hypersurface $x=0$ from which I can integrate my system for an equation of this type $$\Bigl(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\Bigr)f(x,y)=g(x,y)$$
with $$f(0,y)=something$$

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  • $\begingroup$ If g = 0 and 'something' = 0, function f = A sin(x) exp(-y) satisfies both the equation and your initial condition. Therefore such problem has non-unique solution. $\endgroup$ – atarasenko Oct 29 '18 at 14:45

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