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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)
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