In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study.
Such statements give a striking qualitative picture of solutions of the given differential equation.
In many situations, one can also use such maximum principles to draw precise quantitative conclusions about solutions of differential equations, such as control over the size of their gradient.
[2] Here we consider the simplest case, although the same thinking can be extended to more general scenarios.
Fix some choice of x in M. According to the spectral theorem of linear algebra, all eigenvalues of the matrix [aij(x)] are real, and there is an orthonormal basis of ℝn consisting of eigenvectors.
This elementary reasoning could be argued to represent an infinitesimal formulation of the strong maximum principle, which states, under some extra assumptions (such as the continuity of a), that u must be constant if there is a point of M where u is maximized.
Note that the above reasoning is unaffected if one considers the more general partial differential equation since the added term is automatically zero at any hypothetical maximum point.
This phenomenon is important in the formal proof of the classical weak maximum principle.
However, the above reasoning no longer applies if one considers the condition since now the "balancing" condition, as evaluated at a hypothetical maximum point of u, only says that a weighted average of manifestly nonpositive quantities is nonpositive.
This is reflected by any number of concrete examples, such as the fact that and on any open region containing the origin, the function −x2−y2 certainly has a maximum.
is maximized at a point p, then one automatically has: One can view a partial differential equation as the imposition of an algebraic relation between the various derivatives of a function.
So, if u is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of u form a contradiction to this algebraic relation.
Clearly, the applicability of this idea depends strongly on the particular partial differential equation in question.
is a function such that which is a sort of "non-local" differential equation, then the automatic strict positivity of the right-hand side shows, by the same analysis as above, that u cannot attain a maximum value.
For instance, if u is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point p where
However, one could consider, for an arbitrary real number s, the function us defined by It is straightforward to see that By the above analysis, if
All that mattered was to have a function which extends continuously to the boundary and whose Laplacian is strictly positive.
If one can make the choice of h so that the right-hand side has a manifestly positive nature, then this will provide a contradiction to the fact that x0 is a maximum point of u on M, so that its gradient must vanish.
Direct calculation shows There are various conditions under which the right-hand side can be guaranteed to be nonnegative; see the statement of the theorem below.
Lastly, note that the directional derivative of h at x0 along the inward-pointing radial line of the annulus is strictly positive.
As described in the above summary, this will ensure that a directional derivative of u at x0 is nonzero, in contradiction to x0 being a maximum point of u on the open set M. The following is the statement of the theorem in the books of Morrey and Smoller, following the original statement of Hopf (1927): Let M be an open subset of Euclidean space ℝn.
on M, then u does not attain a maximum value on M.The point of the continuity assumption is that continuous functions are bounded on compact sets, the relevant compact set here being the spherical annulus appearing in the proof.
One then takes α, as appearing in the proof, to be large relative to these bounds.
Evans's book has a slightly weaker formulation, in which there is assumed to be a positive number λ which is a lower bound of the eigenvalues of [aij] for all x in M. These continuity assumptions are clearly not the most general possible in order for the proof to work.
For instance, the following is Gilbarg and Trudinger's statement of the theorem, following the same proof: Let M be an open subset of Euclidean space ℝn.
Suppose that for all x in M, the symmetric matrix [aij] is positive-definite, and let λ(x) denote its smallest eigenvalue.
on M, then u does not attain a maximum value on M.One cannot naively extend these statements to the general second-order linear elliptic equation, as already seen in the one-dimensional case.
For instance, the ordinary differential equation y″ + 2y = 0 has sinusoidal solutions, which certainly have interior maxima.
This extends to the higher-dimensional case, where one often has solutions to "eigenfunction" equations Δu + cu = 0 which have interior maxima.