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Nonlinear parabolic variational inequalities. (English) Zbl 0572.35063

The paper is concerned with a nonlinear parabolic variational inequality with a nonlinearity having quadratic growth in the spatial gradient. Let \(\Omega\) be a bounded open subset of \({\mathbb{R}}^ N\) with smooth boundary and set \(Q=]0,T[\times \Omega\). Let \(a_{ij}\in L^{\infty}(Q)\) for \(i,j=1,...,N\) and suppose that \(\{a_{ij}\}\) is uniformly positive definite in Q. Let H be a function, which is measurable in (t,x)\(\in Q\) and continuous in \((u,p)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), such that \(\forall C\in {\mathbb{R}}\) \(\exists K_ 1,K_ 2\in {\mathbb{R}}:\) \(| H(t,x,u,p)| \leq K_ 1+K_ 2| p|^ 2\) whenever (t,x)\(\in Q\), \((u,p)\in {\mathbb{R}}\times {\mathbb{R}}^ N\), \(| u| \leq C\). Finally, let \(\psi\) : \(Q\to {\mathbb{R}}\cup \{-\infty \}\) be a Borel function.
The author studies bounded weak solutions of the nonlinear parabolic variational inequality associated with the equation \[ u_ t- \sum_{i,j}D_{x_ i}a_{ij}D_{x_ j}u+H(t,x,u,D_ xu)=0 \] and the constraint \(u\geq \psi\). First of all a result is proved, which establishes a relation between the oscillation of u and the Wiener modulus of \(\psi\), in order to get informations about the (Hölder) continuity of u.
Then, under suitable assumptions on \(\psi\), the existence of a continuous weak solution with null initial value is proved. Moreover, uniqueness results (under suitable assumptions on H) and a result of further summability for \(D_ xu\) (under suitable assumptions on \(\psi)\) are proved for continuous weak solutions. Finally, a dual inequality and related regularity results are proved under suitable assumptions on \(\psi\).
Reviewer: M.Degiovanni

MSC:

35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35K55 Nonlinear parabolic equations
49J40 Variational inequalities
35B65 Smoothness and regularity of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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