Generalized solution

A generalization of the concept of a classical solution of a differential (pseudo-differential) equation. It arose in relation to many problems in mathematical physics where it was necessary to regard as solutions of differential equations insufficiently differentiable functions, or even nowhere-differentiable functions, or even more general objects such as generalized functions, hyperfunctions, etc. Thus, the concept of a generalized solution is closely related to those of a generalized derivative and a generalized function. The concept of a generalized solution goes back to L. Euler [9].

A generalized solution of the differential equation

$$ \tag{1 } L ( x , D ) ( u) \equiv \ \sum _ {| \alpha | \leq m } a _ \alpha D ^ \alpha u ( x) = f ( x) , $$

$$ f \in D ^ \prime ( O) ,\ a _ \alpha \in C ^ \infty ( O) , $$

in the class $ D ^ \prime ( O) $ is any generalized function $ u $ in $ D ^ \prime ( O) $ satisfying equation (1) in $ O $, that is, for any test function $ \phi \in D ( O) $, the equation $ ( u , L ^ {*} \phi ) = ( f , \phi ) $ must be satisfied. Here $ L ^ {*} $ is the operator adjoint to $ L $ in the sense of Lagrange:

$$ L ^ {*} \phi = \ \sum _ {| \alpha | \leq m } (- 1) ^ {| \alpha | } D ^ \alpha ( a _ \alpha \phi ) . $$

A generalized solution of a boundary value problem should satisfy the boundary condition in the appropriate generalized sense (in $ L _ {p} ( \partial O ) $ or $ D ^ \prime ( \partial O ) $, etc.), for example: $ u ( r s ) \rightarrow u ( s) $, $ r \uparrow 1 $, in $ L _ {2} ( | s | = 1 ) $; $ u ( x , t ) \rightarrow u _ {0} ( x) $, $ t \downarrow 0 $, in $ D ^ \prime $.

Generalized solutions of boundary value problems for differential equations arise when the latter are solved by variational methods, when applying difference methods, and also as weak limits of classical solutions when applying the Fourier method, the limiting-amplitude principle, pseudo-viscosity methods, etc.

Examples.

1) The general solution of the equation $ x ^ {2} u ^ \prime = 0 $ in the class $ D ^ \prime ( \mathbf R ) $ is given by

$$ u ( x) = C _ {1} + C _ {2} \theta ( x) + C _ {3} \delta ( x) , $$

where $ \theta $ is the Heaviside function: $ \theta ( x) = 1 $, $ x \geq 0 $; $ \theta ( x) = 0 $, $ x < 0 $; $ \delta $ is the Dirac delta-function, and $ C _ {1} , C _ {2} \dots $ here and below are arbitrary constants.

2) The equation $ x ^ {2} u ^ \prime + u = 0 $ has a single solution, equal to $ \theta ( - x ) e ^ {1/x} $, in the class $ C ^ \infty ( \mathbf R ) $, but in the class of hyperfunctions its general solution is given by $ u ( x) = C _ {4} e ^ {1 / ( x - i 0 ) } + C _ {5} e ^ {1 / ( x + i 0 ) } + C _ {6} \theta ( - x ) e ^ {1/x} $.

3) The general solution of the wave equation $ u _ {tt} = a ^ {2} u _ {xx} $ in the class $ C ( \mathbf R ^ {2} ) $ is given by $ u ( x , t ) = f ( x + a t) + g ( x - a t ) $, where $ f $ and $ g $ are arbitrary functions of class $ C ( \mathbf R ) $.

4) Every solution $ u $ in $ D ^ \prime ( O) $ of the Laplace equation $ \Delta u = 0 $ is (real) analytic in $ O $.

5) Every solution $ u $ in $ D ^ \prime $ of the heat equation $ u _ {t} = a ^ {2} \Delta u $ is infinitely differentiable.

6) Every differential operator $ L \not\equiv 0 $ with constant coefficients has a fundamental solution of slow growth (in the class $ S ^ \prime $).

7) Every equation $ L ( D) u = f $, where $ L ( D) \not\equiv 0 $ is a differential operator with constant coefficients, has a generalized solution $ u $ in $ L _ {2} ( O) $ for any $ f $ in $ L _ {2} ( O) $, if $ O $ is a bounded domain.

8) A generalized solution $ u $ of the boundary value problem

$$ \tag{2 } \Delta u = f ,\ \ u \mid _ {\partial O } = 0 ,\ \ f \in L _ {2} ( O) , $$

in the Sobolev class $ W _ {2} ^ {(} 1) ( O) $ arises as a solution of the classical variational problem of the minimum of the quadratic functional

$$ J ( u) = \int\limits _ { O } \left ( \sum _ { i= } 1 ^ { n } u _ {x _ {i} } ^ {2} + 2 u f \right ) d x $$

in the class $ W _ {2} ^ {(} 1) ( O) $. The solution of this variational problem exists and is unique in $ W _ {2} ^ {(} 1) ( O) $ for any $ f $ in $ L _ {2} ( O) $. Thus, the generalized solution of (2) gives, for all $ f \in L _ {2} ( O) $, a self-adjoint extension of the operator $ \Delta $( a so-called rigid or Friedrichs extension). The generalized solution of (2) together with its first derivatives are regular in $ O $( that is, are of the type of locally integrable functions on $ O $); its second derivatives are, generally speaking, singular generalized functions.

References

[1]	S.L. Sobolev, "Méthode nouvelle à résoudre le problème de Cauchy pour les équations linéaires hyperboliques normales" Mat. Sb. , 1 : 1 (1936) pp. 39–72
[2]	S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[3]	L. Schwartz, "Théorie des distributions" , 1–2 , Hermann (1950–1951)
[4]	I.M. Gel'fand, G.E. Shilov, "Some problems in differential equations" , Moscow (1958) (In Russian)
[5]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985)
[6]	H. Komatsu (ed.) , Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971 , Lect. notes in math. , 287 , Springer (1973)
[7]	V.S. Vladimirov, "Equations of mathematical physics" , M. Dekker (1971) (Translated from Russian)
[8]	V.S. Vladimirov, "Generalized functions in mathematical physics" , MIR (1979) (Translated from Russian)
[9]	L. Euler, "Institutionum calculi integralis" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 11–13 , Teubner (1913–1914)

Comments

The generalization of the concept of boundary value and boundary conditions requires considerable care in the case of solutions in $ D ^ \prime ( O) $, see e.g. the discussion in [5], Vol. 3, App. B.

Concerning (pseudo-) viscosity methods see also Viscosity solutions.