§ Nonexistence of solutions for ODE and PDE

• ODE system, no bc: always solution by picard liendolf
• ODE system, with boundary cond:, can have no solution. Eg. $f'(x) = 0$, with boundary conditoin $f(a) = 0, f(b) = 1$.
• PDE system, no bc: can still create no solutions!
• PDE system, with boundary cond: can have no solution because ODE is PDE.

§ Example 1 of PDE with no solutions

• Take a vector field on $\mathbb R^2$ with $V(x, y) = (-y, x)$. This vector field has concentric spirals.
• consider this vector field as a PDE, so we are looking for a function $f$such that $\nabla f = V$.
• No such potential function can exist, because this vector field allow us to extract work.
• Suppose such a potential exists. Then if I travel in a circle, according to the potential, net work is zero. But if I evaluate the integral, I will get work done. Thus, no soln exists!
• In general, asking if a differential form is exact is literally asking for a PDE to be solved!
• In this case, the form is also closed , since it's a 2D form on a 2D surface. This is an example of a closed form that is not exact.
• It's nice to see PDE theory and diffgeo connect

§ Example 2: use second axis as time

• Consider a PDE on a square $[0, 1]\times [0, 1]$. We will think of the first axis as space where the function is defined and the second axis as time where the function is perturbed.
• We start by saying $\partial f / \partial x = t$. So the function at $t=0$ is constant, and at $t=1$ is linear.
• Next, we say that $\partial f / partial t = 0$. This means that the function is not allowed to evolve through time.
• This is nonsensical, becase at $t=1$, we expect a constant function to have become a linear function, but along the time axis, we say that no point in space can change.
• Thus, this DE has no solutions!
• We can use the extra dimensions available in a PDE to create "conflicting" data along different time axes.