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

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.

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
PDF
BibTeX
XML
Cite

\textit{M. G. Garroni} and \textit{M. A. Vivaldi}, Nonlinear Anal., Theory Methods Appl. 8, 300--312 (1984; Zbl 0553.35039)

Full Text:
DOI

### References:

[1] | {\scGarroni} M. G. & {\scVivaldi} M. A., Bilateral evolution problems of non-variational type, existence, uniqueness, Hölder-regularity and approximation of solutions, Manuscripta math. (to appear). |

[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 Paris · Zbl 0411.49002 |

[4] | Friedman, A., Stochastic differential equations and applications, Vol. II, (1975), Academic Press New York |

[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. |

[7] | {\scBiroli} M. & {\scMosco} U., Stability and homogenization for nonlinear variational inequalities with irregular obstacles and quadratic growth (to be published). · Zbl 0508.49009 |

[8] | Ciarlet, P.V., The finite element method for elliptic problems, (1978), North-Holland Amsterdam |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.