Differential equation, partial, with singular coefficients

A partial differential equation the coefficients and/or the free term of which have discontinuities of the first kind or become infinite on certain manifolds in the closure of their domain of definition.

Typical equations of this kind are the Lavrent'ev–Bitsadze equation

$$ \mathop{\rm sign} y \cdot u _ {xx} + u _ {yy} = 0 $$

and the Euler–Poisson–Darboux equation

$$ u _ {xy} + \frac{\beta ^ \prime }{y - x } u _ {x} - \frac \beta {y - x } u _ {y} = 0 ,\ \beta , \beta ^ \prime = \textrm{ const } , $$

or

$$ \Delta u - u _ {x _ {0} x _ {0} } = \frac \lambda {x _ {0} } u _ {x _ {0} } , $$

$$ \Delta u - u _ {x _ {0} x _ {0} } = \frac \lambda {x _ {0} } u _ {x _ {0} } ,\ \lambda = \textrm{ const } , $$

where $ \Delta $ is the Laplace operator with respect to the variables $ x _ {1} \dots x _ {n} $.

Many degenerate partial differential equations are equations with singular coefficients.

Of prime importance in the theory of differential equations with singular coefficients is the study of the solvability of initial value, boundary value and mixed problems in their classical and generalized formulations, as well as the search for new well-posed problems. A fairly complete study was made of boundary value problems for linear elliptic, hyperbolic and parabolic equations of the second order with weak singularities in the coefficients that are usually summable of a larger degree than the dimension of their domain of definition. If the coefficients of these equations have discontinuities of the first kind only on certain sufficiently smooth surfaces located within the domain of definition, a fairly complete theory of the principal boundary value problems is available. See Differential equation, partial, discontinuous coefficients, and also [3], [5], [6].

References

[1]	A.V. Bitsadse, "Equations of mixed type" , Pergamon (1964) (Translated from Russian)
[2]	I.N. Vekua, "Generalized analytic functions" , Pergamon (1962) (Translated from Russian) MR0152665 MR0150320 MR0138774 Zbl 0127.03505 Zbl 0100.07603
[3]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian) MR0244627 Zbl 0177.37404 Zbl 0164.13002
[4]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian) MR0284700 Zbl 0198.14101
[5]	O.A. Oleinik, "Solution of fundamental boundary value problem for second-order equations with discontinuous coefficients" Dokl. Akad. Nauk. SSSR , 124 : 6 (1959) pp. 1219–1222 (In Russian) MR0102653
[6]	A.A. Samarskii, "Parabolic equations with discontinuous coefficients" Dokl. Akad. Nauk. SSSR , 121 : 2 (1958) pp. 225–228 (In Russian)

Comments

For non-linear equations singularities of coefficients may occur for particular values of the unknown function. In such a case the set of singular points can be described as a free boundary (see Differential equation, partial, free boundaries). A typical example is

$$ \tag{a1 } \frac \partial {\partial t } C ( u) + \lambda H ( u) = \Delta K ( u) , $$

where $ C ( u) = \int _ {0} ^ {u} c ( v) d v $, $ K ( u) = \int _ {0} ^ {u} k ( v) d v $( $ c $ and $ k $ representing heat capacity and thermal conductivity, respectively), $ \lambda $ is the latent heat, and $ H $ is the Heaviside function. Equation (a1) describes heat conduction with change of phase ( $ u = 0 $ is the melting point). See Stefan problem. Of course, all the derivatives in (a1) are understood in the generalized sense.

The free boundary for (a1) coincides with the set $ \{ u = 0 \} $. If it consists of a smooth surface $ \phi ( x , t ) = 0 $( $ x \in \mathbf R ^ {n} $), then it can be shown that (a1) is equivalent to solving

$$ c ^ \prime ( u) \frac{\partial u }{\partial t } = \ \mathop{\rm div} ( k ( u) \mathop{\rm grad} u ) $$

in the set $ \{ u \neq 0 \} $, with the conditions

$$ u = 0 ,\ [ k ( u) \mathop{\rm grad} u \cdot \mathop{\rm grad} \phi ] _ {-} ^ {+} - \lambda \frac{\partial \phi }{\partial t } = 0 $$

on the free boundary, $ [ \cdot ] _ {-} ^ {+} $ denoting the difference between the limits from the positivity and the negativity set of $ u $.