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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.

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
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References:

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