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A boundary value problem asks for a differential equation solution together with conditions on the boundary of a specified domain. These can prescribe the value, a normal derivative, a flux, or a combination. For an ordinary differential equation on an interval, the conditions are often distributed between its endpoints; a partial differential equation uses the spatial boundary and, for evolution, also needs appropriate initial data.

For the Dirichlet boundary conditions intwice integrating gives $u(x)=x(1-x)/2$. The boundary conditions determine the two constants. By contrast, for $-u^{\prime \prime }=f$ with derivative-zero conditions at both ends, integrating the equation requires ${\int }_0^{1}f=0$ and leaves an additive constant undetermined. This is the Neumann boundary-value problem compatibility and normalization issue.

Existence, uniqueness and continuous dependence on forcing belong to the whole operator/domain/boundary specification. A formal differential formula alone does not establish that a boundary value problem is solvable.^[ref]

Suppose $-u^{\prime \prime }+u+q(u)=f$ and $-v^{\prime \prime }+v+q(v)=g$ on a bounded interval, the solutions have the same Dirichlet boundary conditions, and $q$ is a nondecreasing function. Multiplying the difference equation by $w=u-v$ and using integration by parts givesThe nonlinear integral is nonnegative. Young's inequality for products gives the displayed energy estimate. It proves uniqueness and continuous dependence on forcing. The same argument holds for nodal values when integration by parts is replaced by discrete summation by parts.

Write $w=u-v$ and $r=f-g$. Subtraction of the two boundary value problems gives $-w^{\prime \prime }+w+u^3-v^3=r$, with $w(0)=w(1)=0$. Multiply by $w$ and apply integration by parts:Consequently the required energy identity isThe nonlinear term is nonnegative pointwise, sinceThis is the monotonicity energy estimate for a semilinear Dirichlet problem for the monotone function $s\mapsto s^3$. Apply Young's inequality for products, $rw\le (r^2+w^2)/2$, inside the last integral and discard the nonnegative nonlinear term. Moving $\frac{1}{2}{\int }_0^1{w}^{2}dx$ to the left givesIn particular, equal forcing gives $w=0$, proving uniqueness and a quantitative continuous dependence on forcing estimate for this boundary value problem.