I have been taught to think about initial value problems and boundary conditions as "analogous" in some sense; things that need to be specified in order for a solution to a differential equation to be unique (usually in science classes), but is there a uniqueness and existence theorem that guarantees that there exists only one solution to a Dirichlet problem for, say, a second order differential equation, under some conditions?
I understand that the Piccard-Lindelöf theorem states that for a first order ODE with an initial value $y(x_0)$ there exists only one solution (under reasonable smoothness assumptions), and this can be generalized to a n ODE of order n with $y'(x_0)$, $y''(x_0)$, etc as initial values. My question is whether, for a second order ODE, it is sufficient to specify the values of the function at two points plus, perhaps, some extra conditions.
It seems to me it is not sufficient, because for instance an equation of the form $y'' = -y$ together with the condition $y(0)=0, y(\pi)=0$ admits many solutions, namely the family $y = A\sin(x)$, including $y = 0$, but if one is intelligent enough to choose one of the boundary conditions to be a value that is not $0$ (for instance $y(0)=0, y(\frac{\pi}{2}) =1$) then the problem is indeed well posed, and when working with PDEs boundary conditions seem to often (always?) pick out only one solution.
Best Answer
Solvability condition of a linear ODE $L[y(x)] = f(x)$ with specified boundary conditions on $(a,b)$ is given by $$ \int_a^b u_H^* f(x) \, dx = 0, \label{1} \tag{1}$$ where $u_H^*$ is the homogeneous solution to the adjoint linear operator $L^*$ (with adjoint boundary conditions). If $u_H^* =0$, then the solution exists and is unique, if $u_H^* \neq 0$, then a solution only exists if \eqref{1} is satisfied. The boundary conditions might be a little confusing, look at this for a second-order ODE http://people.cs.uchicago.edu/~lebovitz/Eodesbook/bv.pdf (page 229, bilinear concomitant).