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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.
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
The weak maximum principle, in this setting, says that for any open bounded subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself. Note that both statements are also true for the minimum of u, since -u 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.
Here we consider the simplest case, although the same thinking can be extended to more general scenarios. Let M be an open subset of Euclidean space and let u be a C2 function on M such that
where for each i and j between 1 and n, aij is a function on M with aij = aji.
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. Denote the eigenvalues by λi and the corresponding eigenvectors by vi, for i from 1 to n. Then the differential equation, at the point x, can be rephrased as
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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.
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
The weak maximum principle, in this setting, says that for any open bounded subset M of the domain of u, the maximum of u on the closure of M is achieved on the boundary of M. The strong maximum principle says that, unless u is a constant function, the maximum cannot also be achieved anywhere on M itself. Note that both statements are also true for the minimum of u, since -u 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.
Here we consider the simplest case, although the same thinking can be extended to more general scenarios. Let M be an open subset of Euclidean space and let u be a C2 function on M such that
where for each i and j between 1 and n, aij is a function on M with aij = aji.
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. Denote the eigenvalues by λi and the corresponding eigenvectors by vi, for i from 1 to n. Then the differential equation, at the point x, can be rephrased as