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On one-sided problems for nonlinear operators with convex constraints on the gradient of the solution. (English. Russian original) Zbl 0536.35027
Sov. Math., Dokl. 27, 33-36 (1983); translation from Dokl. Akad. Nauk SSSR 268, 38-41 (1983).
One-sided problems for non-linear elliptic operators of second order with some convex gradient constraints are considered: (1) Lu(x)$$\geq 0$$, $$\nabla u(x)\in K(x)$$, $$x\in \Omega$$; $$\nabla u(x_ 0)\in int K(x_ 0)\Rightarrow Lu(x_ 0)=0$$, $$x_ 0\in \Omega$$; $$u(x)=0$$, $$x\in \partial \Omega$$. Here, $$\Omega \subset {\mathbb{R}}^ n$$ is a bounded domain with smooth boundary, $$L\delta =a^{ij}(x,\nabla \delta)\delta_{x_ ix_ j}+a(x,\delta,\nabla \delta), a^{ij}$$, a satisfy some neutral conditions, $$K(x)\subset {\mathbb{R}}^ n$$ are closed strictly convex sets, 0$$\in int K(x) \forall x\in {\bar \Omega}$$; the sets K(x) satisfy some smoothness conditions and (for unbounded sets K(x)) conditions at infinity.
Some results of solvability and regularity are obtained. Theorem 1 (existence of solution of problem (1) in $$\cap_{1<p<\infty}W^ 2_{p,loc}(\Omega)\cap C^{0,1}({\bar \Omega}))$$ is obtained under the following assumption: $$\exists w_ 1\in C^ 2(\Omega)\cap C^ 1({\bar \Omega}):\quad Lw_ 1(x)\geq 0,\quad \nabla w_ 1(x)\in K(x)$$ in $$\Omega$$, $$w_ 1(x)=0$$ on $$\partial \Omega$$ (lower barrier). To prove this theorem, a regularization of problem (1) is introduced by using some modification of the distance function. A priori estimates are derived, then we have limit passage. By theorem 2 (regularity result) the solution of problem (1) which is obtained in theorem 1, belongs to $$\cap_{1<p<\infty}W^ 2_ p(\Omega)$$ if the solution $$w_ 2$$ of the Dirichlet problem $$Lw_ 2=0$$ in $$\Omega$$, $$w_ 2=0$$ on $$\partial \Omega$$ satisfies the following condition: $$\nabla w_ 2(x)\in K(x)$$, $$\forall x\in \partial \Omega$$. Note that for linear elliptic operators and $$K(x)=\{p\in {\mathbb{R}}^ n:| p| \leq g(x),g\geq 0,g\in C^ 2\}$$ solvability in $$\cap_{1<p<\infty}W^ 2_{p,loc}(\Omega)\cap C^{0,1}({\bar \Omega})$$ of problem (1) has been proved by L. C. Evans [Commun. Partial Differ. Equations 4, 555-572 (1979; Zbl 0448.35036)].

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)