chapter  5
38 Pages

Extremality results for quasilinear PDE

Chapter 5

Extremality results for

quasilinear PDE

The upper and lower solution method combined with the monotone iteration

technique has been proved to be a powerful tool to obtain the existence

of extremal solutions of elliptic and parabolic boundary value problems

within an order interval [u; u] formed by an ordered pair of upper and lower

solutions u and u, respectively. However, in order to apply the monotone

iteration technique to such type of problems given in an abstract form by

Au = Fu; the operator A which stands for an elliptic or parabolic operator

related with some boundary and initial conditions has to have an increasing

inverse, and the operator F which stands for the Nemytskij operator of

the lower order terms has to be increasing with respect to the underlying

natural partial ordering of functions. For example, semilinear elliptic and

parabolic problems with a nonlinear lower order term depending only on

u which is either an increasing or a Lipschitz continuous function of u can

always be transformed by the maximum principle into an abstract setting

given above. Thus the above problem may be rewritten as a xed point

equation

u = A

Fu

within some function space governed by an increasing xed point operator

A

F : [u; u]! [u; u] of the order interval [u; u] to itself which allows one

to obtain the extremal solutions within [u; u] by successive approximation.