On regularity of solutions to inner obstacle problems (Q686389)

The authors discuss the regularity of the solutions \(u\in H^ 1_ 0(\Omega)\) to the inner obstacle problem for variational inequalities \[ \langle Au,v-u\rangle= a(u,v- u)> \langle f,v- u\rangle \] for all \(v\in K_ \psi=\{ v\in H^ 1_ 0(\Omega)\), \(v\geq \psi\) a.e. on \(E\}\), \(E\subset \overline E\subsetneqq \Omega\subset R^ n\), \(\Omega\) be an bounded domain with \(C^{1,1}\)-boundary \(\partial\Omega\), \(E\) open set with \(C^{1,\alpha}\)-boundary \(\partial E= F^{-1}(0)\) for some \(F\in C^{1,\alpha}(R^ n)\), \(a(\cdot,\cdot)\) is given by the formula \[ a(u,v)= \int_ \Omega \bigl\{(a^{ij} D_ i u+ b^ j) D_ j v+ (c^ i D_ i u+ du)v\bigr\} dx\qquad (u,v\in H^ 1_ 0(\Omega)), \] where \(a^{ij}, b^ j\in C^{0,1}(\overline\Omega)\), \(c^ i\), \(d\in L^ \infty(\Omega)\), \(a^{ij} \xi_ i\xi_ \geq \nu|\xi|^ 2\), \(\nu> 0\) and \(a\) is coercive over \(H^ 1_ 0(\Omega)\). Main result is following: If \(\psi\in C^ 0(\overline E)\cap C^ 2(E)\) assumed to satisfy the following assumptions (egg-shaped obstacle) \((A_ 1)\) \(\lim_{x\to x_ 0} D_ i\psi()D_ i F(x)= -\infty\) for every \(x_ 0\in \partial E\), \((A_ 2)\) if vector field \(\vec\tau\in C^{1,\alpha}(\overline E)\) satisfies \(\tau_ i D_ i F= 0\), then \(\tau_ i D_ i\psi\in C^{1,\alpha}(\overline E)\) and \(f\in L^ \infty(\Omega)\) then the solution of the preceding problem belong to \(H^{2,p}_ 0(\Omega)\) for every \(p< \infty\) (and so to \(C^{1,\beta}(\overline\Omega)\) for every \(\beta< 1\)).