In
mathematics, and particularly in
potential theory, Dirichlet's principle is the assumption that the minimizer of a certain
energy functional The energy functional is the total energy of a certain system, as a functional of the system's state.
In the energy methods of simulating the dynamics of complex structures, a state of the system is often described as an element of an appropriat ...
is a solution to
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
.
Formal statement
Dirichlet's principle states that, if the function
is the solution to
Poisson's equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
:
on a
domain of
with
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
:
on the
boundary ,
then ''u'' can be obtained as the minimizer of the
Dirichlet energy
:
amongst all twice differentiable functions
such that
on
(provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician
Peter Gustav Lejeune Dirichlet.
History
The name "Dirichlet's principle" is due to
Riemann, who applied it in the study of
complex analytic functions.
Riemann (and others such as
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
and Dirichlet) knew that Dirichlet's integral is bounded below, which establishes the existence of an
infimum; however, he took for granted the existence of a function that attains the minimum.
Weierstrass published the first criticism of this assumption in 1870, giving an example of a functional that has a greatest lower bound which is not a minimum value. Weierstrass's example was the functional
:
where
is continuous on