Dirichlet boundary condition
As a Dirichlet boundary condition (after Peter Gustav Lejeune Dirichlet ) is called in the context of differential equations (more precisely, boundary value problems ) values to be adopted at the respective edge of the domain of the function. Further boundary conditions are for example Neumann boundary conditions or oblique boundary conditions.
- 2.1 The Dirichlet problem
- 2.2 Example
Ordinary differential equation
The Dirichlet problem
In the case of an ordinary differential equation of the domain of the function is a closed interval. Consequently, there is the edge of the definition range of only the right and left end interval. Due to the freedom in ordinary differential equations Dirichlet boundary conditions are only useful for equations of second or higher order. In this case looks a Dirichlet problem, that is, a differential equation with Dirichlet boundary condition, as follows:
This is a prescribed function, and are prescribed real numbers for the function values of a solution on the interval ends. Finally, we look for a (classical ) solution from the specified Regularitätsklasse.
Example
We choose as our interval and consider the following Dirichlet problem:
With the theory of linear ordinary differential equations with constant coefficients, we first get a general (classical ) solution of the differential equation:
With two arbitrary real constants and. We use the boundary conditions to fix these constants. We obtain a linear system of equations in the unknowns and:
Remarkably, this system is not uniquely solvable, but there is a solution given for arbitrary real by
Existence and uniqueness
The following theorem is formulated for homogeneous data (). However, this is not a limitation, because by a transformation
Can be converted into a problem homogenens an inhomogeneous problem always.
Consider the task
It should be a continuous function. Moreover, it satisfies a Lipschitz condition, that is, there are numbers such that for all and for all the inequality
Was fulfilled. continues to apply
Let be a solution of
Disappear for and is the first unique number such that for. Then the underlying object has exactly one solution if
Applies the other hand, so there must be no solution or it need not be unique. Furthermore, applies
A proof of this theorem can be found in Bailey, Shampine, Waltman. Nonlinear two- point boundary value problems. Academic Press, 1968.
If the right side of the equation, however, only continuous and bounded, then the set of Scorza Dragoni guarantees the existence of a solution.
Partial Differential Equations
The Dirichlet problem
For a partial differential equation, the sole indication of Neumann boundary conditions is only useful for elliptic equations on a bounded domain, since the other types also require specifications of the initial values. This Dirichlet boundary conditions are prescribed on the boundary of the domain. Here, we define the Dirichlet problem for a quasilinear partial differential equation:
Here, the function of the prescribed function values of our solution dar. alone the question of the solvability of such a problem is very challenging and is the focus of current research. It is also very difficult to give a general solution method.
Example
We consider in this example, in the field the following boundary value problem:
Herein, the Laplace operator. First, we note that a solution of the problem is. We want to find more solutions. We assume henceforth for and make the following product approach
For the functions we derive ordinary differential equations with appropriate Dirichlet boundary conditions here. It follows
Now if the the boundary value problem
Sufficient, then the function defined above is a solution of the Dirichlet boundary value problem for the partial differential equation. With the example of ordinary differential equations, we obtain
And thus
As a solution to our problem of partial differential equations for Dirichlet boundary conditions. It remains an open question whether there are other solutions.