Approximation results for bilateral nonlinear problems of nonvariational type. (English) Zbl 0553.35039

Garroni, Maria Giovanna; Vivaldi, Maria Agostina

Nonlinear Anal., Theory Methods Appl. 8, 300-312 (1984).

Some approximation results are given for bilateral evolution problems for linear and quasi-linear second order operators whose principal part is not in divergence form. For a linear operator E ”regularizing” operators \(E^ n\) are introduced whose coefficients are smooth and converge to the coefficients of E. Due to the regularity of the coefficients, the operators \(E^ n\) can be written in divergence form. The order of convergence of strong solutions \(u^ n\) of variational regularizing problems to the strong solution u of the given problem is studied: this is done in terms of the order of convergence of the coefficients. As is well-known only problems with ”very regular” obstactles admit strong solutions: so approximations have been extended to generalized solutions in the case of Hölder-continuous obstacles in another paper by the authors. The case of a quasilinear operator E-G where G is a Nemytsky operator associated with a continuous function g(x,t,u,p) with quadratic growth in the gradient variable p is also considered. Some examples of operators \(G^ n\) converging to G are given.
Finally this estimate can be used in order to study the convergence of free boundaries of variational, regularizing, problems to the free boundary of the given problem: this is shown, by the authors, in a forthcoming paper.

MSC:

35G10	Initial value problems for linear higher-order PDEs
35K25	Higher-order parabolic equations
35K60	Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35A35	Theoretical approximation in context of PDEs
35K85	Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators

Keywords:

operators not in divergence form; regularizing operators; approximation results; bilateral evolution problems; obstactles; quasilinear operator; convergence of free boundaries

References:

[2]	Troianiello, G. M., On solutions to quasi linear parabolic unilateral problems, Boll. Un. mat. ital., 6, 535-552 (1982) · Zbl 0527.35047
[3]	Bensoussan, A.; Lions, J. L., Applications des Inéquations Variationnelles en Contrôle Stochastique (1978), Dunod: Dunod Paris · Zbl 0411.49002
[4]	Friedman, A., Stochastic Differential Equations and Applications, Vol. II (1975), Academic Press: Academic Press New York · Zbl 0323.60056
[5]	Garroni, M. G.; Vivaldi, M. A., Bilateral inequalities and implicit unilateral systems of the non-variational type, Manuscripta math., 33, 177-215 (1980) · Zbl 0452.35043
[6]	Ladyzenskaya, O. A.; Solonnikov, V. A.; Ural’ceva, N. N., Linear and quasilinear equations of parabolic type, Am. Math. Soc. Transl., 23 (1968), Providence, R. I. · Zbl 0174.15403
[8]	Ciarlet, P. V., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058