# Differential equation, partial, discontinuous initial (boundary) conditions

A problem involving partial differential equations in which the functions specifying the initial (boundary) conditions are not continuous.

For instance, consider the second-order hyperbolic equation

$$\frac{\partial ^ {2} u }{\partial t ^ {2} } = a ^ {2} \frac{\partial ^ {2} u }{\partial x ^ {2} } + f ,\ 0 \langle x < 1 ,\ t \rangle t _ {0} ,$$

and pose for it the mixed problem with initial conditions

$$\left . \frac{\partial u }{\partial t } \right | _ {t = t _ {0} } = \phi _ {1} ,\ \left . u \right | _ {t = t _ {0} } = \phi _ {0} ,$$

and boundary conditions

$$\left . u \right | _ {x=} 0 = \psi _ {1} ,\ \left . u \right | _ {x=} 1 = \psi _ {2} .$$

In this case the discontinuities of the initial functions $\phi _ {0}$ and $\phi _ {1}$ entail discontinuities of $u$ and $\partial u / \partial t$ along the characteristic rays $x - at = \textrm{ const }$ and $x + at = \textrm{ const }$, and the measure of discontinuity

$$\chi = u ( c \pm at + 0 , t ) - u ( c \pm at - 0 , t ) ,$$

or

$$\chi = u _ {t} ( c \pm at + 0 , t ) - u _ {t} ( c \pm at - 0 , t ) ,$$

where $c \in [ 0 , 1 ]$ is a discontinuity point of the function $\phi _ {0}$ or $\phi _ {1}$, satisfies the equation

$$\frac{d \chi }{dt } + 0 \cdot \chi = 0$$

along the characteristic ray, i.e. $\chi = \textrm{ const }$. Similar results are valid for second-order hyperbolic equations with variable coefficients:

$$\frac{\partial ^ {2} u }{\partial t ^ {2} } = \ \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ( x) \frac{\partial u }{\partial x _ {i} } + c ( x) u + f ,$$

$$\left . u \right | _ {t = t _ {0} } = \phi _ {0} ( x) ,\ \left . u _ {t} \right | _ {t = t _ {0} } = \phi _ {1} ( x) ,\ \left . u \right | _ {\partial D } = \psi .$$

In this case the discontinuities of the initial functions and the boundary conditions also entail discontinuities in $u$ and $\partial u / \partial t$ along characteristic rays, which can be determined from the systems of equations

$$\frac{d x _ {i} }{dt } = \sum _ {j = 1 } ^ { n } a _ {ij} ( x) \phi _ {j} ,\ \sum _ {i , j = 1 } ^ { n } a _ {ij} ( x) \phi _ {i} \phi _ {j} = 0 .$$

The measure of discontinuity $\chi$ satisfies the equation:

$$2 \frac{d \chi }{dt } + A \chi = 0 , \ A = \sum _ {i , j = 1 } ^ { n } a _ {ij} \frac{\partial ^ {2} \phi }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} \frac{\partial \phi }{ \partial x _ {i} } ,$$

where the function $\phi ( x)$ defines the characteristic surface in the form of the equation $\phi ( x) = C$.

In the case of equations of elliptic type the discontinuities of the boundary conditions do not propagate inside $D$ because in this case the characteristic rays are complex. For equations of elliptic type studies were made of the existence and uniqueness of the solution, and of the solution satisfying the boundary conditions. Thus, it has been proved for second-order elliptic equations in an arbitrary domain,

$$\sum _ {i , j = 1 } ^ { n } a _ {ij} ( x) \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum _ {i = 1 } ^ { n } b _ {i} ( x) \frac{\partial u }{\partial x _ {i} } + c ( x) u = f ,$$

$$\left . u \right | _ {\partial D } = \psi \ \textrm{ or } \ \frac{\partial u }{\partial n } + k ( x) \left . u \right | _ {\partial D } = \psi ,$$

that if the boundary function $\psi \in W _ {2} ^ {1/2} ( \partial D )$ for the first boundary condition and $\psi \in L _ {2} ( \partial D )$ for the second boundary condition, then there exists a generalized solution in $W _ {2} ^ {1} ( D)$ which satisfies the boundary condition on the average, i.e. $\| u - \psi \| _ {L _ {2} ( \partial D _ {n} ) } \rightarrow 0$, where the surfaces $\partial D _ {n}$ approximate the surface $\partial D$. In the case of parabolic (and also elliptic) equations, the discontinuities do not propagate inside $D$ if discontinuities are present in the initial or in the boundary conditions. Problems of the existence and uniqueness of a generalized solution to the boundary condition have also been studied for these problems.

#### References

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