Differential equation, partial, Fischer-Riesz (Picone) method

A method for solving boundary value problems of partial differential equations based on the use of the Green formula and leading to a system of (Fischer–Riesz) integral equations for some suitably chosen unknown vector. The method may be used to find numerical values of solutions, but may also be employed in proving existence theorems.

Let $ L ^ {*} $ and $ L $ be adjoint linear elliptic operators of the second order in $ \mathbf R ^ {n} $ with real coefficients

$$ a _ {ik} \in C ^ {( 2) } ( D) \cap C ^ {( 1) } ( D \cup S ) , $$

$$ b _ {i} \in C ^ {( 1) } ( D \cup S ) ,\ c , f \in C ^ {( 0) } ( D \cup S ) , $$

where $ D $ is the bounded domain bounded by a closed surface $ S $. Let the solution $ u $ of the Dirichlet problem

$$ L u ( x) = f ( x) ,\ x \in D ,\ \lim\limits _ {x \rightarrow y \in S } u ( x) = \phi ( y) , $$

be sought in the class of functions permitting an integral representation according to Green's formulas. Furthermore, let $ v $ be an arbitrary function in the same class. Application of the Green formula to $ u $ and $ v $ yields

$$ \tag{1 } - \int\limits _ { D } u ( x) L ^ {*} v ( x) d x = $$

$$ = \ \int\limits _ { S } \left [ \phi ( y) \left ( \frac{\partial v }{\partial \nu } - A v \right ) - v \frac{\partial u }{\partial \nu } \right ] dy - \int\limits _ { D } f ( x) v ( x) d x , $$

where

$$ A = \sum _ { i } \cos ( n , x _ {i} ) \left ( b _ {i} - \sum _ { k } \frac{\partial a _ {ik} }{\partial x _ {k} } \right ) , $$

$ n $ is a normal on $ S $ and $ \nu $ is the interior conormal. Let $ U = ( u _ {1} , u _ {2} ) $ be a vector with two components, composed of real-valued square-integrable functions, the first component being defined in $ D $, while the second is defined on $ S $. Let $ L _ {2} ( D , S ) $ be the set of these vectors; a norm is introduced by way of the scalar product of $ U $ and $ V $ in $ L _ {2} ( D , S ) $:

$$ ( U , V ) = \int\limits _ { D } u _ {1} v _ {1} d x + \int\limits _ { S } u _ {2} v _ {2} d y . $$

Let $ \{ v _ {k} \} $ be a set which has been so chosen that the totality of vectors with two components

$$ V _ {k} = ( - L ^ {*} v _ {k} ( x) , v _ {k} ( y) ) ,\ \ x \in D ,\ y \in S , $$

is dense in the Hilbert space $ L _ {2} ( D , S ) $. Then, if one denotes by $ U = ( u _ {1} , u _ {2} ) $ the vector with first component $ u _ {1} $ equal to $ u $, and second component $ u _ {2} $ coinciding with $ \partial u / \partial \nu $, one may write (1) as a Fischer–Riesz system of integral equations:

$$ \tag{2 } ( U , V _ {k} ) = \int\limits _ { S } \phi ( y) \left ( \frac{\partial v _ {k} }{\partial \nu } - A v _ {k} \right ) dy - \int\limits _ { D } f v _ {k} d x . $$

If the set $ \{ V _ {k} \} $ is orthonormal and if the conditions of the Riesz–Fischer theorem are satisfied, (2) defines, in $ L _ {2} ( D , S ) $, the Fourier coefficients $ c _ {k} $ of the vector $ U = ( u _ {1} , u _ {2} ) $ with respect to the maximal system of basis vectors $ \{ V _ {k} \} $. If it is known that the problem under consideration has a solution $ v $ and that this solution is unique, the Fourier series $ \sum _ {k} c _ {k} V _ {k} $ converges in the mean to $ v $ and only to $ v $. Otherwise the selection of functions $ \{ v _ {k} \} $ must be further studied. E.g., if eigen solutions $ u _ {0} $ are permitted (that is, the solution is no longer unique), the set $ \{ V _ {k} \} $ must satisfy:

$$ ( U _ {0} , V _ {k} ) = 0 , $$

where $ U _ {0} = ( u _ {0} ( x) , \partial u _ {0} / \partial \nu ) $.

If a sequence of monomials $ x _ {1} ^ {\alpha _ {1} } \dots x _ {m} ^ {\alpha _ {m} } $ with integral non-negative exponents $ \alpha _ {1} \dots \alpha _ {m} $ is taken as $ \{ v _ {k} \} $, then the values of $ u $ and $ \partial u / \partial \nu $ found by (2), together with the value of $ u $ given on $ S $, satisfy the Green functional relations

$$ \delta ( x) k _ {m} u ( x) = $$

$$ = \ \int\limits _ { S } \left [ u ( y) \left ( \frac{\partial w ( y , x ) }{\partial \nu } - A ( y) w ( y , x ) \right ) - w ( y , x ) \frac{\partial u }{\partial \nu } \right ] d y + $$

$$ - \int\limits _ { D } f w ( y , x ) d y , $$

$$ \delta ( x) = \left \{ \begin{array}{ll} 1 , & x \in D , \\ 0, & x \notin D \cup S , \\ \end{array} \right .$$

where $ k _ {m} $ is a non-zero constant which depends on $ m $, and $ w $ is a fundamental solution of the equation $ L ^ {*} v = 0 $. In such a case all solutions of the Fischer–Riesz system of equations, and only such solutions, are solutions of the boundary value problem under study. The essence of this method is a suitable construction of the selected set of functions $ \{ v _ {n} \} $ satisfying the condition $ L ^ {*} v _ {n} = 0 $ or certain completeness conditions [4].

In this method, an explicit expression for the fundamental solution need not be specified, but if it is known, the calculations may be considerably simplified in view of the fact that the set $ \{ w ( y , x ^ {( k) } ) \} $, where $ x ^ {( k) } $ is a countably infinite sequence of arbitrary points not forming part of $ D \cup S $, is linearly independent and is complete in $ L _ {2} ( S) $[4]; this theorem also makes it possible to extend the method of Fischer–Riesz equations to problems with oblique derivatives (cf. Differential equation, partial, oblique derivatives) and other types of equations.

References

[1]	M. Picone, "Nuovi metodi resolutivi per i problemi d'integrazzione delle equazioni lineari a derivative parziali e nuova applicazione della trasformata multipla di Laplace nel caso delle equazioni a coefficienti constanti" Atti Accad. Sci. Torino. Cl. Sci. Fis. Mat. Natur. , 75 (1940) pp. 413–426
[2]	L. Amerio, "Sul calcolo delle soluzioni dei problemi al contorno per le equazioni lineari de secondo ordine di tipo ellittico" Amer. J. Math. , 69 : 3 (1947) pp. 447–489
[3]	G. Fichera, "Sull' esistenza e sul calcolo delle soluzioni dei problemi al contorno, relativi all'equilibrio di un corpo elastico" Ann. Scuola Norm. Super. Pisa. Sci. Fis. Mat. , 4 : 1–2 (1950) pp. 35–99
[4]	V.D. Kupradze, "On the approximate solution of problems in mathematical physics" Russian Math. Surveys , 22 : 2 (1967) pp. 58–108 Uspekhi Mat. Nauk , 22 : 2 (1967) pp. 59–107
[5]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)