Maximum principle

A partial formulation of the strong maximum principle Here we consider the simplest case, although the same thinking can be extended to more general scenarios. Let be an open subset of Euclidean space and let be a function on such that :\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}=0 where for each and between 1 and , is a function on with . Fix some choice of in . According to the spectral theorem of linear algebra, all eigenvalues of the matrix are real, and there is an orthonormal basis of consisting of eigenvectors. Denote the eigenvalues by and the corresponding eigenvectors by , for from 1 to . Then the differential equation, at the point , can be rephrased as :\sum_{i=1}^n \lambda_i \left. \frac{d^2}{dt^2}\right|_{t=0}\big(u(x+tv_i)\big)=0. The essence of the maximum principle is the simple observation that if each eigenvalue is positive (which amounts to a certain formulation of "ellipticity" of the differential equation) then the above equation imposes a certain balancing of the directional second derivatives of the solution. In particular, if one of the directional second derivatives is negative, then another must be positive. At a hypothetical point where is maximized, all directional second derivatives are automatically nonpositive, and the "balancing" represented by the above equation then requires all directional second derivatives to be identically zero. 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 ), that must be constant if there is a point of where is maximized. Note that the above reasoning is unaffected if one considers the more general partial differential equation :\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i \, \partial x^j}+\sum_{i=1}^n b_i\frac{\partial u}{\partial x^i}=0, since the added term is automatically zero at any hypothetical maximum point. The reasoning is also unaffected if one considers the more general condition :\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i \, \partial x^j}+\sum_{i=1}^n b_i\frac{\partial u}{\partial x^i}\geq 0, in which one can even note the extra phenomena of having an outright contradiction if there is a strict inequality ( rather than ) in this condition at the hypothetical maximum point. This phenomenon is important in the formal proof of the classical weak maximum principle. Non-applicability of the strong maximum principle However, the above reasoning no longer applies if one considers the condition :\sum_{i=1}^n\sum_{j=1}^n a_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+\sum_{i=1}^n b_i\frac{\partial u}{\partial x^i}\leq 0, since now the "balancing" condition, as evaluated at a hypothetical maximum point of , only says that a weighted average of manifestly nonpositive quantities is nonpositive. This is trivially true, and so one cannot draw any nontrivial conclusion from it. This is reflected by any number of concrete examples, such as the fact that :\frac{\partial^2}{\partial x^2}\big({-x}^2-y^2\big)+\frac{\partial^2}{\partial y^2}\big({-x}^2-y^2\big)\leq 0, and on any open region containing the origin, the function certainly has a maximum. == The classical weak maximum principle for linear elliptic PDE ==

The essential idea Let denote an open subset of Euclidean space. If a smooth function u:M\to\mathbb{R} is maximized at a point , then one automatically has: • (du)(p)=0 • (\nabla^2 u)(p)\leq 0, as a matrix inequality. One can view a partial differential equation as the imposition of an algebraic relation between the various derivatives of a function. So, if is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of form a contradiction to this algebraic relation. This is the essence of the maximum principle. Clearly, the applicability of this idea depends strongly on the particular partial differential equation in question. For instance, if solves the differential equation :\Delta u=|du|^2+2, then it is clearly impossible to have \Delta u\leq 0 and du=0 at any point of the domain. So, following the above observation, it is impossible for to take on a maximum value. If, instead solved the differential equation \Delta u=|du|^2 then one would not have such a contradiction, and the analysis given so far does not imply anything interesting. If solved the differential equation \Delta u=|du|^2-2, then the same analysis would show that cannot take on a minimum value. The possibility of such analysis is not even limited to partial differential equations. For instance, if u:M\to\mathbb{R} is a function such that :\Delta u-|du|^4=\int_M e^{u(x)}\,dx, 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 cannot attain a maximum value. There are many methods to extend the applicability of this kind of analysis in various ways. For instance, if is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point where \Delta u(p)\leq 0 is not in contradiction to the requirement \Delta u=0 everywhere. However, one could consider, for an arbitrary real number , the function defined by :u_s(x)=u(x)+se^{x_1}. It is straightforward to see that :\Delta u_s=se^{x_1}. By the above analysis, if s>0 then cannot attain a maximum value. One might wish to consider the limit as to 0 in order to conclude that also cannot attain a maximum value. However, it is possible for the pointwise limit of a sequence of functions without maxima to have a maxima. Nonetheless, if has a boundary such that together with its boundary is compact, then supposing that can be continuously extended to the boundary, it follows immediately that both and attain a maximum value on M\cup\partial M. Since we have shown that , as a function on , does not have a maximum, it follows that the maximum point of , for any , is on \partial M. By the sequential compactness of \partial M, it follows that the maximum of is attained on \partial M. This is the weak maximum principle for harmonic functions. This does not, by itself, rule out the possibility that the maximum of is also attained somewhere on . That is the content of the "strong maximum principle," which requires further analysis. The use of the specific function e^{x_1} above was very inessential. All that mattered was to have a function which extends continuously to the boundary and whose Laplacian is strictly positive. So we could have used, for instance, :u_s(x)=u(x)+s|x|^2 with the same effect. ==The classical strong maximum principle for linear elliptic PDE==

Summary of proof Let be an open subset of Euclidean space. Let u:M\to\mathbb{R} be a twice-differentiable function which attains its maximum value . Suppose that :a_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+b_i\frac{\partial u}{\partial x^i}\geq 0. Suppose that one can find (or prove the existence of): • a compact subset of , with nonempty interior, such that for all in the interior of , and such that there exists on the boundary of with . • a continuous function h:\Omega\to\mathbb{R} which is twice-differentiable on the interior of and with ::a_{ij}\frac{\partial^2h}{\partial x^i\,\partial x^j}+b_i\frac{\partial h}{\partial x^i}\geq 0, : and such that one has on the boundary of with Then on with on the boundary of ; according to the weak maximum principle, one has on . This can be reorganized to say :-\frac{u(x)-u(x_0)}\geq \frac{h(x)-h(x_0)} for all in . If one can make the choice of so that the right-hand side has a manifestly positive nature, then this will provide a contradiction to the fact that is a maximum point of on , so that its gradient must vanish. Proof The above "program" can be carried out. Choose to be a spherical annulus; one selects its center to be a point closer to the closed set than to the closed set , and the outer radius is selected to be the distance from this center to ; let be a point on this latter set which realizes the distance. The inner radius is arbitrary. Define :h(x)=\varepsilon\Big(e^{-\alpha|x-x_{\text{c}}|^2}-e^{-\alpha R^2}\Big). Now the boundary of consists of two spheres; on the outer sphere, one has ; due to the selection of , one has on this sphere, and so holds on this part of the boundary, together with the requirement . On the inner sphere, one has . Due to the continuity of and the compactness of the inner sphere, one can select such that . Since is constant on this inner sphere, one can select such that on the inner sphere, and hence on the entire boundary of . Direct calculation shows :\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2h}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial h}{\partial x^i}=\varepsilon \alpha e^{-\alpha|x-x_{\text{c}}|^2}\left(4\alpha\sum_{i=1}^n\sum_{j=1}^n a_{ij}(x)\big(x^i-x_{\text{c}}^i\big)\big(x^j-x_{\text{c}}^j\big)-2\sum_{i=1}^n a_{ii}-2 \sum_{i=1}^n b_i\big(x^i-x_{\text{c}}^i\big)\right). 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 at 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 at is nonzero, in contradiction to being a maximum point of on the open set . Statement of the theorem The following is the statement of the theorem in the books of Morrey and Smoller, following the original statement of Hopf (1927): {{blockquote|Let be an open subset of Euclidean space . For each and between 1 and , let and be continuous functions on with . Suppose that for all in , the symmetric matrix is positive-definite. If is a nonconstant function on such that :\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial u}{\partial x^i}\geq 0 on , then does not attain a maximum value on .}} 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. Furthermore, by the same principle, there is a number such that for all in the annulus, the matrix has all eigenvalues greater than or equal to . One then takes , 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 which is a lower bound of the eigenvalues of for all in . 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: {{blockquote|Let be an open subset of Euclidean space . For each and between 1 and , let and be functions on with . Suppose that for all in , the symmetric matrix is positive-definite, and let denote its smallest eigenvalue. Suppose that \textstyle\frac{a_{ii}}{\lambda} and \textstyle\frac{\lambda} are bounded functions on for each between 1 and . If is a nonconstant function on such that :\sum_{i=1}^n\sum_{j=1}^na_{ij}\frac{\partial^2u}{\partial x^i\,\partial x^j}+\sum_{i=1}^nb_i\frac{\partial u}{\partial x^i}\geq 0 on , then does not attain a maximum value on .}} 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 has sinusoidal solutions, which certainly have interior maxima. This extends to the higher-dimensional case, where one often has solutions to "eigenfunction" equations which have interior maxima. The sign of c is relevant, as also seen in the one-dimensional case; for instance the solutions to are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions. == See also ==