View source for 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|generalized derivative]] and a [[Generalized function|generalized function]]. The concept of a generalized solution goes back to L. Euler [[#References|[9]]].

A generalized solution of the differential equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438802.png" /></td> </tr></table>

in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438803.png" /> is any generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438804.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438805.png" /> satisfying equation (1) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438806.png" />, that is, for any test function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438807.png" />, the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438808.png" /> must be satisfied. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g0438809.png" /> is the operator adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388010.png" /> in the sense of Lagrange:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388011.png" /></td> </tr></table>

A generalized solution of a boundary value problem should satisfy the boundary condition in the appropriate generalized sense (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388012.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388013.png" />, etc.), for example: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388015.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388016.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388018.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388019.png" />.

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|Fourier method]], the [[Limiting-amplitude principle|limiting-amplitude principle]], pseudo-viscosity methods, etc.

===Examples.===


1) The general solution of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388020.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388021.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388022.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388023.png" /> is the Heaviside function: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388025.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388027.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388028.png" /> is the Dirac [[Delta-function|delta-function]], and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388029.png" /> here and below are arbitrary constants.

2) The equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388030.png" /> has a single solution, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388031.png" />, in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388032.png" />, but in the class of hyperfunctions its general solution is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388033.png" />.

3) The general solution of the wave equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388034.png" /> in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388035.png" /> is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388038.png" /> are arbitrary functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388039.png" />.

4) Every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388041.png" /> of the [[Laplace equation|Laplace equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388042.png" /> is (real) analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388043.png" />.

5) Every solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388044.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388045.png" /> of the [[Heat equation|heat equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388046.png" /> is infinitely differentiable.

6) Every differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388047.png" /> with constant coefficients has a [[Fundamental solution|fundamental solution]] of slow growth (in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388048.png" />).

7) Every equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388049.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388050.png" /> is a differential operator with constant coefficients, has a generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388052.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388054.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388055.png" /> is a bounded domain.

8) A generalized solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388056.png" /> of the boundary value problem

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

in the Sobolev class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388058.png" /> arises as a solution of the classical variational problem of the minimum of the quadratic functional

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in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388060.png" />. The solution of this variational problem exists and is unique in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388061.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388063.png" />. Thus, the generalized solution of (2) gives, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388064.png" />, a self-adjoint extension of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388065.png" /> (a so-called rigid or Friedrichs extension). The generalized solution of (2) together with its first derivatives are regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388066.png" /> (that is, are of the type of locally integrable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388067.png" />); its second derivatives are, generally speaking, singular generalized functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.L. Sobolev,   "Applications of functional analysis in mathematical physics" , Amer. Math. Soc.  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''1–2''' , Hermann  (1950–1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Some problems in differential equations" , Moscow  (1958)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1–4''' , Springer  (1983–1985)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer  (1973)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.S. Vladimirov,   "Equations of mathematical physics" , M. Dekker  (1971)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  L. Euler,   "Institutionum calculi integralis"  G. Kowalewski (ed.) , ''Opera Omnia Ser. 1; opera mat.'' , '''11–13''' , Teubner  (1913–1914)</TD></TR></table>



====Comments====
The generalization of the concept of boundary value and boundary conditions requires considerable care in the case of solutions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043880/g04388068.png" />, see e.g. the discussion in [[#References|[5]]], Vol. 3, App. B.

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