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If you have the value of a function of many variables, and its 1st-derivatives, at a single point, and a 2nd-order partial differential equation, then haven't you determined the entire function? You can use a Taylor expansion about that point to build the entire function because you have the value of the function at that point, the value of the 1st derivatives, and the value of higher derivatives (from the differential equation and differentiation of the differential equation).
Cauchy boundary conditions require the value of a function and the value of the normal derivative to an entire boundary curve of the region of interest. But don't you just need these things at a single point, and not an entire boundary curve?
Cauchy boundary conditions require the value of a function and the value of the normal derivative to an entire boundary curve of the region of interest. But don't you just need these things at a single point, and not an entire boundary curve?