Upper-and-lower-functions method

A method for demonstrating the existence of solutions of boundary value problems for differential equations. The idea of this method applied to ordinary differential equations was discussed in the work of G. Peano (1880). For the case of the Dirichlet problem and for the case of the Laplace equation the idea occurs in H. Poincaré's balayage method. O. Perron [1] was the first to give a full exposition of the method of upper and lower functions for this last case.

Let the Dirichlet problem be posed in a region $ G $ of the space $ \mathbf R ^ {n} $, $ n \geq 2 $, for a linear, homogeneous, elliptic second-order equation with continuous coefficients,

$$ \tag{1 } Lu \equiv \sum_{i,j=1} ^ { n } a _ {ij} \frac{\partial ^ {2} u }{\partial x _ {i} \partial x _ {j} } + \sum_{i=1} ^ { n } b _ {i} \frac{\partial u }{\partial x _ {i} } + cu = 0, $$

$$ c \leq 0,\ x \in G, $$

with the boundary condition

$$ \tag{2 } u ( x) = f ( x),\ x \in \partial G. $$

In the method of upper and lower functions one introduces a generalization of superharmonic functions (respectively, of subharmonic functions), under the assumption that the problems (1) and (2) are locally solvable. A function $ v $ which is continuous in $ G $ is said to be a generalized superharmonic function (respectively, a generalized subharmonic function) in $ G $ if for any sufficiently small ball $ K $, $ \overline{K}\; \subset G $, the inequality $ ( v) _ {K} \leq v $( respectively, $ ( v) _ {K} \geq v $) is true. Here, $ ( v) _ {K} $ is a continuous function in $ G $ which is equal to $ v $ outside $ K $ and on its boundary and which satisfies equation (1) inside $ K $. For a continuous function $ f $ on the boundary $ \partial G $, a generalized superharmonic (respectively, a subharmonic) function $ v $ is said to be upper (respectively, lower) if for $ x \in \partial G $ the inequality $ v( x) \geq f( x) $( respectively, $ v( x) \leq f( x) $) is true.

The classes $ \Phi ( G, f ) $ and $ \Psi ( G, f ) $ of all upper and lower functions, respectively, are non-empty, and if $ v \in \Phi ( G, f ) $ and $ w \in \Psi ( G, f ) $, then $ v \geq w $[3]. A generalized solution of the Dirichlet problem is defined as the smallest envelope of the class $ \Phi ( G, f ) $ or as the largest envelope of the class $ \Psi ( G, f ) $:

$$ \tag{3 } u ( x) = \inf \{ {v ( x) } : {v \in \Phi ( G, f ) } \} = $$

$$ = \ \sup \{ w ( x): w \in \Psi ( G, f ) \} ,\ x \in G. $$

If the boundary $ \partial G $ permits a barrier at each one of its points, then $ u( x) = f( x) $ everywhere on $ \partial G $, i.e. $ u $ is a classical solution of the Dirichlet problem. In the general case the behaviour of the generalized solution (3) of the elliptic equation (1) at boundary points fully parallels the behaviour of the generalized solution of the Laplace problem (cf. Perron method).

The method of upper and lower functions is also employed in the study of the first boundary value problem for linear homogeneous parabolic second-order equations of the form

$$ Lu - \frac{\partial u }{\partial t } = 0,\ \ ( x, t) \in G \times [ 0, T], $$

with the initial condition

$$ u ( x, 0) = f ( x, 0),\ \ x \in G, $$

and boundary condition

$$ u ( x, t) = f ( x, t),\ \ ( x, t) \in \partial G \times [ 0, T]. $$

In this case superparabolic (subparabolic) functions, with properties analogous to those of generalized superharmonic (subharmonic) functions, are introduced [4].

References

[1]	O. Perron, "Eine neue Behandlung der ersten Randwertaufgabe für $\Delta u = 0$" Math. Z. , 18 (1923) pp. 42–54
[2]	I.G. Petrovskii, "Partial differential equations" , Saunders (1967) (Translated from Russian)
[3]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German)
[4]	V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)

Comments

In axiomatic potential theory, this method is usually called the Perron–Wiener–Brelot method. In its general form, this method defines the upper solution of the Dirichlet problem, for any open set $ \Omega $ satisfying the minimum principle and any numerical boundary function $ f $, as the greatest lower bound of all upper functions: the lower-bounded hyperharmonic functions $ u $ on $ \Omega $ such that $ \lim\limits \inf u \geq f $ on the boundary and $ u \geq 0 $ outside some compact set. The lower solution is defined in a dual way. The Dirichlet problem admits a generalized solution if the upper and lower solutions are equal and harmonic (i.e. satisfy the differential equation). In this case, the boundary function is said to be resolutive. See [a1].

References

[a1]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)