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Nonlinear parabolic variational inequalities. (English) Zbl 0653.49009

Integral functionals in calculus of variations, Proc. Int. Workshop, Trieste/Italy 1985, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 15, 181-188 (1987).
[For the entire collection see Zbl 0636.00006.]
This paper deals with parabolic variational inequalities (V.I.) for second order quasilinear operators with quadratic growth in the spatial gradient. This type of problems comes e.g. from the stochastic control theory by the dynamic programming approach.
The V.I. considered presents difficulties due to the quadratic nonlinearity of the operators as well as difficulties typical of evolutionary unilateral problems.
Existence results for weak and strong solutions are established by using quite different assumptions and methods.
For the irregular obstacle case, the existence of weak solutions is proved by an approximation procedure, using penalty methods, monotonicity techniques, comparison arguments and showing, for the approximate solutions, uniform estimates. When the obstacle and the coefficients of the operators are “regular” regularity results in Sobolev spaces are proved: the main devices are a bootstrap argument, interpolation inequalities between Hölder spaces and Sobolev spaces, Meyers type estimates and a dual estimate of Lewy-Stampacchia type. Finally, the existence of strong solutions for a larger class of operators (and more regular obstacles) is proved by a variant of the Schauder-Tychonov fixed point theorem, since comparison and monotonicity tools don’t seem to work for this type of approximate problems.
Reviewer: M.A.Vivaldi

MSC:

49J40 Variational inequalities
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35D05 Existence of generalized solutions of PDE (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47H05 Monotone operators and generalizations
47H10 Fixed-point theorems

Citations:

Zbl 0636.00006