Differential equation, partial, functional methods

functional-analytic methods for partial differential equations

Methods in which the left-hand side is considered as an operator acting on a suitably defined function space. The best developed functional-analytic methods concern linear partial differential equations. In such a case functional-analytic methods may be divided in two classes: a) methods concerning the spectral theory of differential operators; and b) methods used to clarify the general aspects of the solvability of equations, boundary value problems and properties of solutions. Class b) is in turn conveniently subdivided into methods based exclusively on general theorems of functional analysis (the theory of operators), and methods involving the techniques of Fourier transformation. The former involve the direct study of boundary value problems for general equations with variable coefficients; in the latter, the starting point is the study of differential operators with constant coefficients, often unconnected with the boundary conditions, while the transition to variable coefficients, if feasible, is effected with the aid of a suitable "perturbation theory" . The results obtained by using the two classes of methods complement each other.

Functional-analytic methods first arose in the context of the connection between the problem of minimization of the functional

$$ \tag{1 } J ( u) = \int\limits _ { V } \left \{ \left ( \frac{\partial u }{\partial x } \right ) ^ {2} + \left ( \frac{\partial u }{\partial y } \right ) ^ {2} - 2 f u \right \} d V $$

in the class of real-valued functions $ u $ which on the boundary $ S $ of a domain $ V $( taken as two-dimensional, for the sake of simplicity) satisfy the condition

$$ \tag{2 } \left . u \right | _ {S} = 0 , $$

and the problem of solving the Poisson equation

$$ \tag{3 } - \Delta u = f $$

in $ V $( equation (3) is the Euler equation for (1) under the same boundary conditions (2)). If $ f $ is not smooth (e.g. $ f \in H ( V) $, where $ H ( V) $ is the Hilbert space of square-integrable functions in $ V $), then a minimum for (1) is attained on functions which, generally speaking, do not have the second derivatives required in equation (3), i.e. these functions merely yield a generalized solution of the problem (3), (2). An exhaustive description of this situation [1], [2] involves the Hilbert space $ {W ^ { {down } 60 \circ } } {} ^ {1} $ obtained by completion of the linear manifold of smooth functions which satisfy (2) in the norm induced by the scalar product

$$ \{ u , v \} = \int\limits _ { V } \left \{ \frac{\partial u }{\partial x } \frac{\partial v }{\partial x } + \frac{\partial u }{\partial y } \frac{\partial v }{\partial y } \right \} d V . $$

A solution of the problem (3), (2) is then defined as an element $ u \in {W ^ { {down } 60 \circ } } {} ^ {1} $ such that the following equation is satisfied for any element $ v \in {W ^ { {down } 60 \circ } } {} ^ {1} $:

$$ \tag{4 } \{ u , v \} = ( f , v ) , $$

where the round brackets denote the scalar product in $ H ( V) $. For any $ f \in H $ the minimum of (1) is also attained in $ {W ^ { {down } 60 \circ } } {} ^ {1} $, and the minimum yields a unique solution of the problem (3), (2) in the sense (4). Any classical (i.e. having second derivatives) solution of the problem (3), (2) satisfies equations (4) for any permissible function $ v $, which means that it is a generalized solution. This entails, in particular, the definition by means of generalized solutions of a certain extension of the classical operator $ - \Delta $, which was originally defined on smooth functions satisfying condition (2). The determination of conditions to be imposed on $ f $ and $ S $ for the generalized solution to be a classical solution is much more involved; this is a problem of the so-called differentiability properties of the generalized solution.

In the scheme discussed above the existence of a generalized solution was a consequence of the existence in $ {W ^ { {down } 60 \circ } } {} ^ {1} $ of an element realizing the minimum of the functional (1). It was subsequently noted that the existence theorem may be obtained directly from definition (4) with the aid of the so-called Riesz theorem on the general form of a bounded linear functional [1]. The subsequent development of the theory proceeds in several directions. The given constructions are immediately generalized to the case when $ - \Delta $ is replaced by a general self-adjoint positive elliptic operator of order $ 2m $ with variable coefficients. Clearly, the scalar product will then contain derivatives of order $ m $. The property of compactness of the inverse operator makes it possible to establish the nature of the spectrum of these problems, and to establish the applicability of the Fredholm alternative. It is also possible to obtain existence of a solution if the equation contains lower-order terms (i.e. in addition to $ - \Delta $ there are also the terms $ a ( x , y ) \{ \partial u / \partial x \} + b ( x , y ) \{ \partial u / \partial y \} $), i.e. without any relation to variational problems [3].

Generalizations of a different kind follow from the truth of the existence and uniqueness theorem of the generalized solution of the problem (3), (2) if $ f $ is an arbitrary bounded linear functional on $ {W ^ { {down } 60 \circ } } {} ^ {1} $. The space of all such functionals may be obtained by completion of $ H $ in the new norm

$$ \tag{5 } | f | _ { {W } ^ {-} 1 } = \sup _ {w \in {W ^ { {down } 60 \circ } } {} ^ {1} } \frac{| ( f , w ) | }{| w | _ { {W ^ { {down } 60 \circ } } {} ^ {1} } } . $$

The space $ {W } ^ {-} 1 $ obtained is much larger than $ H $. For instance, in the one-dimensional case ( $ V $ is the interval $ ( - 1 , 1 ) $), the sequence of functions

$$ \delta _ \epsilon ( x) = \left \{ \begin{array}{ll} \frac{1}{2 \epsilon } , & | x | \leq \epsilon , \\ 0, &\ | x | > \epsilon , \epsilon = 1 , 2 ^ {-} 1 , 2 ^ {-} 2 \dots \\ \end{array} \right .$$

converges in $ {W } ^ {-} 1 $, but diverges in $ H $. The limit of this sequence is not a function in the ordinary sense of the word, i.e. the space $ {W } ^ {-} 1 $ contains generalized functions. At the same time the inequality

$$ | \langle f , w \rangle | \leq | f | _ { {W } ^ {-} 1 } | w | _ { {W ^ { {down } 60 \circ } } {} ^ {1} } $$

(the brackets $ \langle , \rangle $ denote the action of the functional $ f $ on an element $ w \in {W ^ { {down } 60 \circ } } {} ^ {1} $) ensure the validity of the existence and uniqueness theorems in the case $ f \in {W } ^ {-} 1 $. This result is also interesting because the operator $ - \Delta $ now establishes an isomorphism between the spaces $ {W } ^ {-} 1 $ and $ {W ^ { {down } 60 \circ } } {} ^ {1} $. If $ f \in H $ and if $ S $ is smooth, the generalized solution of the problem (3), (2) not only belongs to $ {W ^ { {down } 60 \circ } } {} ^ {1} $, but also has second-order generalized derivatives, so that $ u \in W ^ {2} $, and

$$ | u | _ {W ^ {2} } \leq c | f | _ {H} . $$

If $ n = 2 $( $ n $ is the dimension of $ V $), the space $ W ^ {-} 2 $( of functionals on $ W ^ {2} $) contains the $ \delta $- function, i.e. the functional defined by the equation $ \langle \delta _ {x _ {0} } , u \rangle = u ( x _ {0} ) $. There exists in the space $ H $ a generalized solution of the equation

$$ - \Delta v _ {x _ {0} } = \delta _ {x _ {0} } $$

such that the formula

$$ ( - \Delta u , v _ {x _ {0} } ) = < u , - \Delta v _ {x _ {0} } > = \langle u , \delta _ {x _ {0} } \rangle = u ( x _ {0} ) , $$

or

$$ u ( x _ {0} ) = ( f , v _ {x _ {0} } ) , $$

is valid for the generalized solution of the problem (3), (2). This is an abstract existence theorem for the Green function for the problem (3), (2). The corresponding constructions are also realizable in a space of arbitrary dimension for general elliptic operators [4].

Next to the elliptic operators, general functional-analytic methods were applied successfully to equations with a distinguished "time" variable; those classes of equations generalize the classical parabolic and hyperbolic equations of mathematical physics. In this case the basic results may be arbitrarily subdivided into three classes: those concerning operator equations with first-order and second-order derivatives with respect to time [5]; symmetric positive systems of partial differential equations of the first order [3], [6]; and the Cauchy problem for hyperbolic equations of arbitrary order. The first class of results is usually interpreted either in the context of ordinary differential equations in a Banach space and of the theory of operator semi-groups or, together with the second and third classes, in the framework of the so-called technique of energy inequalities. This technique may be outlined as follows. For smooth solutions of an equation involving time derivatives, written in operator form

$$ \tag{6 } L u = f , $$

subject to appropriate initial and boundary conditions, the following inequality is established:

$$ \tag{7 } | u | _ {H _ {1} } \leq c | L u | _ {H _ {2} } , $$

where $ H _ {1} $ and $ H _ {2} $ are suitably defined function spaces. For the sake of simplicity, they will be considered to be Hilbert spaces. For instance, in the case of a function $ u \in C ^ {2} $ which satisfies the equation

$$ L u \equiv \frac{\partial ^ {2} u }{\partial t ^ {2} } - \frac{\partial ^ {2} u }{\partial x ^ {2} } = f $$

and the conditions

$$ \left . u \right | _ {t=} 0 = \left . u _ {t} ^ \prime \right | _ {t=} 0 = \left . u \right | _ {x=} 0 = \left . u \right | _ {x=} 1 = 0 , $$

in the unit square $ V $, multiplication by $ \partial u / \partial t $, integration with respect to $ V $ and elementary transformation will yield the inequality

$$ \tag{8 } \int\limits _ { t= } T \left [ \left ( \frac{\partial u }{\partial x } \right ) ^ {2} + \left ( \frac{\partial u }{\partial t } \right ) ^ {2} \right ] dx \leq 4 \int\limits _ { V } f ^ {2} dx dt ,\ 0 \leq T \leq 1 . $$

Equation (8) is a mathematical expression of a conservation law, whence the term "energy inequality" , which is equivalent to the inequality (7) if the norms in $ H _ {1} $, $ H _ {2} $ are suitably defined. If one defines a generalized solution of equation (6) as an element $ u \in H _ {1} $ for which there exists a sequence of smooth functions $ u _ {i} $ satisfying the boundary conditions described above, such that $ u _ {i} \rightarrow u $ in $ H _ {1} $ and $ Lu _ {i} \rightarrow f $ in $ H _ {2} $, one obtains at the same time a definition of the operator $ L : H _ {1} \rightarrow H _ {2} $ for which there exists a bounded inverse operator. This is equivalent to the uniqueness theorem of the generalized solution and again yields an extension of the classical definitions of $ L $. The next step in the study of equation (6) is to prove existence of such a solution for any permissible right-hand side $ f \in H _ {2} $. Since the range $ R _ {L} $ of $ L $ is a closed subspace in $ H _ {2} $, the proof of existence is equivalent to the proof that the orthogonal complement to $ R _ {L} $ is empty: From the equation

$$ \tag{9 } ( Lu , v ) _ {2} = 0 \ \textrm{ for any } u \in \mathfrak U _ {L} $$

( $ \mathfrak U $ is the domain of $ L $ in $ H _ {1} $ and $ ( \cdot , \cdot ) _ {2} $ is the scalar product in $ H _ {2} $) implies that $ v = 0 $. From the point of view of the theory of operators, equation (9) means that $ v \in \mathfrak U _ {L ^ {*} } $ and $ L ^ {*} v = 0 $, i.e. the desired result would immediately follow from the uniqueness of a solution of the equation $ L ^ {*} v = g $. However, the proof of such a result would require special constructions, which are often quite complicated, since it is not at all obvious that an inequality analogous to (7) (and a sequence of smooth functions satisfying the required boundary conditions and converging to $ v $) exists for $ L ^ {*} $. (The latter is defined classically by an integral identity.) The resulting difficulties reflect the fact that "time" problems are not self-adjoint. In the study of hyperbolic equations of second and higher order, equation (9) is usually employed to prove the equality $ v = 0 $ as follows: One selects an operator $ P ^ {*} $ such that $ P ^ {*} v \in \mathfrak U _ {L} $ and

$$ \tag{10 } ( L P ^ {*} v , v ) _ {2} \geq c | v | _ {H _ {2} } ^ {2} . $$

In addition, (7) is usually derived from an expression such as $ ( Lu , Q u ) _ {2} $, where $ Q $ is an operator chosen so that

$$ ( L u , Q u ) _ {2} \geq c | u | _ {H _ {1} } ^ {2} . $$

Writing (10) in the form

$$ ( L ^ {*} v , P ^ {*} v ) _ {2} \geq c | v | _ {H _ {2} } ^ {2} , $$

one may conclude that the study of (6) is based on these two inequalities (sometimes called dual). The above mainly applies to hyperbolic equations; parabolic equations can be treated in a manner greatly resembling the case of elliptic equations.

The Cauchy problem for equations with constant coefficients [10] can be exhaustively studied by means of Fourier transformation. Moreover, Fourier transformation, in conjunction with so-called "freezing of coefficients of a partial differential operatorfreezing of coefficients" (i.e. locally replacing an operator with variable coefficients by one with constant coefficients equal to the values of the respective coefficients at some given point), is often employed to obtain energy inequalities using the schemes mentioned above. This also plays an important role in the local characterization of operators and boundary conditions for elliptic equations [10]. Subsequent development of these ideas led to the introduction and investigation of pseudo-differential operators (cf. Pseudo-differential operator). It was also established, by means of Fourier transformation, that the inequality

$$ \tag{11 } | u | _ {H(} V) \leq c | L u | _ {H(} V) $$

applies to all differential operators $ L $ with constant coefficients, defined in a compact domain $ V $ of a Euclidean space and acting on functions $ u \in C _ {0} ^ \infty ( V ) $, i.e. functions which, together with all derivatives, vanish on the boundary. If one takes, for $ L $, the closure $ \dot{L} $ in $ H( V) $( $ C _ {0} ^ \infty ( V) $ being taken as the domain of definition of $ L $), and defines $ \widetilde{L} $ as the adjoint to $ \dot{L} $( in $ H( V) $), one obtains $ R _ {\widetilde{L} } = H( V) $. The existence of the inequality (11) for solutions of the equation (6) ensures that the solution is unique, and this automatically entails the solvability of the equation $ L ^ {*} v = g $ involving the adjoint operator, for any right-hand side. By virtue of Banach's theorem one may conclude that there exists an operator $ {\widehat{L} } $, $ \dot{L} \subset {\widehat{L} } \subset \widetilde{L} $( i.e. $ {\widehat{L} } $ is an extension of the minimal operator $ \dot{L} $ and a restriction of the maximal operator $ \widetilde{L} $) such that $ R _ {\widehat{L} } = H( V) $, and that a bounded inverse operator $ {\widehat{L} } {} ^ {-} 1 $( on $ H $) exists [10]. To some extent $ {\widehat{L} } $ corresponds to a boundary value problem (which is defined by a set of conditions "concentrated" on the boundary "between" the identically zero conditions and the complete absence of conditions). Little is known about the nature of the conditions determining the operator $ \widehat{L} $ in the general case.

The use of the Fourier transformation also serves as the base for investigating a wide range of problems in the theory of differential operators connected with the study of the structure of the fundamental solution and of the so-called local properties of solutions (as distinct from the study of the properties of solutions of boundary value problems). This includes the theorem on the existence of a fundamental solution for a general operator with constant coefficients, the theorem of local (in a neighbourhood of a given point) solvability of the equation with a right-hand side and the smoothness properties of this solution [9], [11]. Since the inhomogeneous equation (6) is not always solvable in the case of variable (complex) coefficients (for the equation

$$ - i \frac{\partial u }{\partial x _ {1} } + \frac{\partial u }{\partial x _ {2} } - 2 ( x _ {1} + i x _ {2} ) \frac{\partial u }{\partial x _ {3} } = f $$

there is no solution for $ f \in C ^ \infty $, even in the generalized sense, in any open non-empty subset of $ \mathbf R ^ {3} $[9]), it is necessary to study conditions insuring local solvability.

The use of functional-analytic methods is very fruitful in the study of non-linear partial differential equations as well. This applies, first and foremost, to variational methods (if a more general functional than (1) is considered, the corresponding Euler equation may be non-linear) [12], to variants of the fixed-point method (cf. Schauder method), to the continuation method [3], [13], and to other methods mainly applied to elliptic and parabolic equations. The method of energy inequalities is successfully employed in the study of quasi-linear hyperbolic equations.

References

[1]	S.L. Sobolev, "Applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963) (Translated from Russian)
[2]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[3]	L. Bers, F. John, M. Schechter, "Partial differential equations" , Interscience (1964)
[4]	J.L. Lions, E. Magenes, "Non-homogenous boundary value problems and applications" , 1–2 , Springer (1972) (Translated from French)
[5]	E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[6]	M. Nagumo, "Lectures on the modern theory of partial differential equations" , Kyôritsu (1957) (In Japanese)
[7]	L. Gårding, "Cauchy's problem for hyperbolic equations" , Third Congress Scand. Math. Helsinki, 1957 , Mercators Trycken' (1958) pp. 104–109
[8]	I.M. Gel'fand, G.E. Shilov, "Some problems in the theory of differential equations" , Moscow (1958) (In Russian)
[9]	L. Hörmander, "Linear partial differential operators" , Springer (1969)
[10]	L. Hörmander, "On the theory of general partial differential operators" Acta Math. , 94 (1955) pp. 161–248
[11]	J. Trèves, "Linear partial differential equations with constant coefficients" , Gordon & Breach (1966)
[12]	O.A. Ladyzhenskaya, N.N. Ural'tseva, "Linear and quasilinear elliptic equations" , Acad. Press (1968) (Translated from Russian)
[13]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)

Comments

The state-of-the-art of functional-analytic methods for linear partial differential operators is well documented in [a3]. Results for the constant-coefficient case are given in Vol. II. Energy integral methods are touched upon in Vol. III. Pseudo-differential operators have proved fruitful for elliptic operators, also with non-elliptic boundary conditions, cf. Vol. III. A later development, Fourier integral operators, can be used for more general problems, see Vol. IV.

References

[a1]	K. Yosida, "Functional analysis" , Springer (1980)
[a2]	F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Gauthier-Villars (1968)
[a3]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985)