Maximum principle

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. Solutions of a partial differential equation (or, more generally, of a differential inequality) in a domain D are said to satisfy the maximum principle if they achieve their maxima at the boundary of D. Harmonic functions and, more generally, solutions of elliptic partial differential equations satisfy the maximum principle.

File:Maximum modulus principle.png

The graph (red) of a typical function in two dimension satisfying the maximum principle: maxima (and minimal) occur on the boundary of the domain (blue).

The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations.

In a simple two-dimensional case, consider a function of two variables u(x,y) such that

$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0.$

The weak maximum principle, in this setting, says that for any open bounded subset Template:Mvar of the domain of Template:Mvar, the maximum of Template:Mvar on the closure of Template:Mvar is achieved on the boundary of Template:Mvar. The strong maximum principle says that, unless Template:Mvar is a constant function, the maximum cannot also be achieved anywhere on Template:Mvar itself. Note that both statements are also true for the minimum of Template:Mvar, since -Template:Mvar solves the same differential equation.

In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary.

Intuition

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 Template:Mvar be an open subset of Euclidean space and let Template:Mvar be a C² function on Template:Mvar such that

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}=0$

where for each Template:Mvar and Template:Mvar between 1 and Template:Mvar, a_ij is a function on Template:Mvar with a_ij = a_ji.

Fix some choice of Template:Mvar in Template:Mvar. According to the spectral theorem of linear algebra, all eigenvalues of the matrix [a_ij(x)] are real, and there is an orthonormal basis of ℝⁿ consisting of eigenvectors. Denote the eigenvalues by λ_i and the corresponding eigenvectors by v_i, for Template:Mvar from 1 to Template:Mvar. Then the differential equation, at the point Template:Mvar, can be rephrased as

$\sum _{i=1}^n{\lambda }_i{\frac{d^2}{d{t}^2}|}_{t=0}(u(x+t{v}_i))=0.$

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 Template:Mvar 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 Template:Mvar), that Template:Mvar must be constant if there is a point of Template:Mvar where Template:Mvar is maximized.

Note that the above reasoning is unaffected if one considers the more general partial differential equation

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial u}{\partial x^i}=0,$

since the added term is automatically zero at any hypothetical maximum point. The reasoning is also unaffected if one considers the more general condition

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial u}{\partial x^i}\ge 0,$

in which one can even note the extra phenomena of having an outright contradiction if there is a strict inequality (Template:Mvar rather than Template:Mvar) 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

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial u}{\partial x^i}\le 0,$

since now the "balancing" condition, as evaluated at a hypothetical maximum point of Template:Mvar, 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

$\frac{{\partial }^2}{\partial x^2}({-x}^2-y^2)+\frac{{\partial }^2}{\partial y^2}({-x}^2-y^2)\le 0,$

and on any open region containing the origin, the function −x²−y² certainly has a maximum.

The classical weak maximum principle for linear elliptic PDE

The essential idea

Let Template:Mvar denote an open subset of Euclidean space. If a smooth function $u:M\to \mathbb{ℝ}$ is maximized at a point Template:Mvar, then one automatically has:

• $(du)(p)=0$
• $({\mathrm{\nabla }}^{2}u)(p)\le 0,$ 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 Template:Mvar is the solution of a partial differential equation, then it is possible that the above conditions on the first and second derivatives of Template:Mvar 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 Template:Mvar solves the differential equation

$\mathrm{\Delta }u=|du{|}^2+2,$

then it is clearly impossible to have $\mathrm{\Delta }u\le 0$ and $du=0$ at any point of the domain. So, following the above observation, it is impossible for Template:Mvar to take on a maximum value. If, instead Template:Mvar solved the differential equation $\mathrm{\Delta }u=|du{|}^2$ then one would not have such a contradiction, and the analysis given so far does not imply anything interesting. If Template:Mvar solved the differential equation $\mathrm{\Delta }u=|du{|}^2-2,$ then the same analysis would show that Template:Mvar 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

$\mathrm{\Delta }u-|du{|}^4=\underset{M}{\int }{e}^{u(x)}dx,$

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 Template:Mvar cannot attain a maximum value.

There are many methods to extend the applicability of this kind of analysis in various ways. For instance, if Template:Mvar is a harmonic function, then the above sort of contradiction does not directly occur, since the existence of a point Template:Mvar where $\mathrm{\Delta }u(p)\le 0$ is not in contradiction to the requirement $\mathrm{\Delta }u=0$ everywhere. However, one could consider, for an arbitrary real number Template:Mvar, the function u_s defined by

$u_s(x)=u(x)+s{e}^{x_1}.$

It is straightforward to see that

$\mathrm{\Delta }{u}_s=s{e}^{x_1}.$

By the above analysis, if $s>0$ then u_s cannot attain a maximum value. One might wish to consider the limit as Template:Mvar to 0 in order to conclude that Template:Mvar 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 Template:Mvar has a boundary such that Template:Mvar together with its boundary is compact, then supposing that Template:Mvar can be continuously extended to the boundary, it follows immediately that both Template:Mvar and u_s attain a maximum value on $M\cup \partial M.$ Since we have shown that u_s, as a function on Template:Mvar, does not have a maximum, it follows that the maximum point of u_s, for any Template:Mvar, is on $\partial M.$ By the sequential compactness of $\partial M,$ it follows that the maximum of Template:Mvar 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 Template:Mvar is also attained somewhere on Template:Mvar. 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 Template:Mvar be an open subset of Euclidean space. Let $u:M\to \mathbb{ℝ}$ be a twice-differentiable function which attains its maximum value Template:Mvar. Suppose that

$a_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+b_i\frac{\partial u}{\partial x^i}\ge 0.$

Suppose that one can find (or prove the existence of):

• a compact subset Template:Mvar of Template:Mvar, with nonempty interior, such that u(x) < C for all Template:Mvar in the interior of Template:Mvar, and such that there exists x₀ on the boundary of Template:Mvar with u(x₀) = C.
• a continuous function $h:\mathrm{\Omega }\to \mathbb{ℝ}$ which is twice-differentiable on the interior of Template:Mvar and with

$a_{ij}\frac{{\partial }^{2}h}{\partial x^i\partial x^j}+b_i\frac{\partial h}{\partial x^i}\ge 0,$

and such that one has u + h ≤ C on the boundary of Template:Mvar with h(x₀) = 0

Then L(u + h − C) ≥ 0 on Template:Mvar with u + h − C ≤ 0 on the boundary of Template:Mvar; according to the weak maximum principle, one has u + h − C ≤ 0 on Template:Mvar. This can be reorganized to say

$-\frac{u(x)-u(x_0)}{|x-x_0|}\ge \frac{h(x)-h(x_0)}{|x-x_0|}$

for all Template:Mvar in Template:Mvar. If one can make the choice of Template:Mvar 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 Template:Mvar on Template:Mvar, so that its gradient must vanish.

Proof

The above "program" can be carried out. Choose Template:Mvar to be a spherical annulus; one selects its center x_c to be a point closer to the closed set u⁻¹(C) than to the closed set ∂M, and the outer radius Template:Mvar is selected to be the distance from this center to u⁻¹(C); let x₀ be a point on this latter set which realizes the distance. The inner radius Template:Mvar is arbitrary. Define

$h(x)=\epsilon (e^{-\alpha |x-x_{\text{c}}{|}^2}-e^{-\alpha R^2}).$

Now the boundary of Template:Mvar consists of two spheres; on the outer sphere, one has h = 0; due to the selection of Template:Mvar, one has u ≤ C on this sphere, and so u + h − C ≤ 0 holds on this part of the boundary, together with the requirement h(x₀) = 0. On the inner sphere, one has u < C. Due to the continuity of Template:Mvar and the compactness of the inner sphere, one can select δ > 0 such that u + δ < C. Since Template:Mvar is constant on this inner sphere, one can select ε > 0 such that u + h ≤ C on the inner sphere, and hence on the entire boundary of Template:Mvar.

Direct calculation shows

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}h}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial h}{\partial x^i}=\epsilon \alpha e^{-\alpha |x-x_{\text{c}}{|}^2}(4\alpha \sum _{i=1}^n\sum _{j=1}^n{a}_{ij}(x)(x^i-x_{\text{c}}^i\left)\right(x^j-x_{\text{c}}^j)-2\sum _{i=1}^n{a}_{ii}-2\sum _{i=1}^n{b}_i(x^i-x_{\text{c}}^i\left)\right).$

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 Template:Mvar 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 Template:Mvar at x₀ is nonzero, in contradiction to x₀ being a maximum point of Template:Mvar on the open set Template:Mvar.

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):

Let Template:Mvar be an open subset of Euclidean space ℝⁿ. For each Template:Mvar and Template:Mvar between 1 and Template:Mvar, let a_ij and b_i be continuous functions on Template:Mvar with a_ij = a_ji. Suppose that for all Template:Mvar in Template:Mvar, the symmetric matrix [a_ij] is positive-definite. If Template:Mvar is a nonconstant C² function on Template:Mvar such that

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial u}{\partial x^i}\ge 0$

on Template:Mvar, then Template:Mvar does not attain a maximum value on Template:Mvar.

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 Template:Mvar such that for all Template:Mvar in the annulus, the matrix [a_ij(x)] has all eigenvalues greater than or equal to Template:Mvar. One then takes Template:Mvar, 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 Template:Mvar which is a lower bound of the eigenvalues of [a_ij] for all Template:Mvar in Template:Mvar.

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 Template:Mvar be an open subset of Euclidean space ℝⁿ. For each Template:Mvar and Template:Mvar between 1 and Template:Mvar, let a_ij and b_i be functions on Template:Mvar with a_ij = a_ji. Suppose that for all Template:Mvar in Template:Mvar, the symmetric matrix [a_ij] is positive-definite, and let λ(x) denote its smallest eigenvalue. Suppose that

$\frac{a_{ii}}{\lambda }$

and

$\frac{|b_i|}{\lambda }$

are bounded functions on Template:Mvar for each Template:Mvar between 1 and Template:Mvar. If Template:Mvar is a nonconstant C² function on Template:Mvar such that

$\sum _{i=1}^n\sum _{j=1}^n{a}_{ij}\frac{{\partial }^{2}u}{\partial x^i\partial x^j}+\sum _{i=1}^n{b}_i\frac{\partial u}{\partial x^i}\ge 0$

on Template:Mvar, then Template:Mvar does not attain a maximum value on Template:Mvar.

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 yTemplate:'' + 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. The sign of c is relevant, as also seen in the one-dimensional case; for instance the solutions to yTemplate:'' - 2y = 0 are exponentials, and the character of the maxima of such functions is quite different from that of sinusoidal functions.

See also

• Maximum modulus principle
• Hopf maximum principle

References

Research articles

• Calabi, E. An extension of E. Hopf's maximum principle with an application to Riemannian geometry. Duke Math. J. 25 (1958), 45–56.
• Cheng, S.Y.; Yau, S.T. Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28 (1975), no. 3, 333–354.
• Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209–243.
• Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in Rⁿ. Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
• Hamilton, Richard S. Four-manifolds with positive curvature operator. J. Differential Geom. 24 (1986), no. 2, 153–179.
• Hopf, Eberhard. Elementare Bemerkungen Über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitber. Preuss. Akad. Wiss. Berlin 19 (1927), 147–152.
• Hopf, Eberhard. A remark on linear elliptic differential equations of second order. Proc. Amer. Math. Soc. 3 (1952), 791–793.
• Nirenberg, Louis. A strong maximum principle for parabolic equations. Comm. Pure Appl. Math. 6 (1953), 167–177.
• Omori, Hideki. Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19 (1967), 205–214.
• Yau, Shing Tung. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975), 201–228.
• Kreyberg, H. J. A. On the maximum principle of optimal control in economic processes, 1969 (Trondheim, NTH, Sosialøkonomisk institutt https://www.worldcat.org/title/on-the-maximum-principle-of-optimal-control-in-economic-processes/oclc/23714026)

Textbooks

• Evans, Lawrence C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. Template:ISBN
• Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp.
• Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. Template:ISBN
• Ladyženskaja, O. A.; Solonnikov, V. A.; Uralʹceva, N. N. Linear and quasilinear equations of parabolic type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968 xi+648 pp.
• Ladyzhenskaya, Olga A.; Ural'tseva, Nina N. Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis. Academic Press, New York-London 1968 xviii+495 pp.
• Lieberman, Gary M. Second order parabolic differential equations. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xii+439 pp. Template:ISBN
• Morrey, Charles B., Jr. Multiple integrals in the calculus of variations. Reprint of the 1966 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2008. x+506 pp. Template:ISBN
• Protter, Murray H.; Weinberger, Hans F. Maximum principles in differential equations. Corrected reprint of the 1967 original. Springer-Verlag, New York, 1984. x+261 pp. Template:ISBN
• Smoller, Joel. Shock waves and reaction-diffusion equations. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258. Springer-Verlag, New York, 1994. xxiv+632 pp. Template:ISBN