| dbp:proof
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- For having simpler formulas, we set
The first part results by using the chain rule for differentiating both sides of the equation with respect to and taking the limit of the result when tends to .
The converse is proved by integrating a simple differential equation.
Let be in the interior of the domain of . For sufficiently close to , the function
is well defined. The partial differential equation implies that
The solutions of this linear differential equation have the form
Therefore, if is sufficiently close to . If this solution of the partial differential equation would not be defined for all positive , then the functional equation would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree . (en)
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