A homogeneity improvement approach to the obstacle problem
Theorem 1. Let \(K= \{k: \mathbb{R}^n\to \mathbb{R}\), \(k(x)= x\cdot Ax\), where \(x\in\mathbb{R}^n\) and \(A\) is nonnegative symmetric marix such that \(\text{tr }A= 1/4\}\), \(\text{ri}K= \{k\in K: k(x)> 0\) if \(x\in \mathbb{R}^n\setminus\{0\}\}\), \(P: H^{1,2}(S(0,1))\to K\) the projection onto \(K\) in \(H^{1,2}(S(0,1))\), \(H= \{x\mapsto{1\over 4}\max(x-\nu,0)^2: \nu\in\partial S(0,1))\}\) and \(M: H^{1,2}(S(0,1))\to \mathbb{R}\), \(M(v)= \int_{S(0,1)}(|\nabla v|^2+ \max(v, 0)) dx- 2\int_{\partial S(0,1)} v^2 dH^{n- 1}\), where \(H^{n- 1}\) is the \((n-1)\)-dimensional Hausdorff measure. Then there exist \(\kappa,\delta\in ]0,1[\) such that it holds for each nonnegative \(c\in H^{1,2}(S(0,1))\) that is homogeneous of degree 2: (1) if \(|c-h|_{1,2}\leq \delta\) for some \(h\in H\), then there exists \(v\in H^{1,2}(S(0,1))\) such that \(v= c\) on \(\partial S(0, 1)\) and \(M(v)\leq (1-\kappa)M(c)+ \kappa{1\over 16} \int_{S(0,1)} x^2_1 dx\), (2) if \(\text{dist}(c, K)\leq\delta\) and either \(P(c)\in \text{ri }K\) or \(P(c)\circ U(x)= {1\over 4} x^2_n\) for all \(x\in \mathbb{R}^n\) for some rotation \(U\), then there exists \(v\in H^{1,2}(S(0,1))\) such that \(v= c\) on \(\partial S(0,1)\) and \(M(v)\leq (1- \kappa)M(c)+ \kappa{1\over 8} \int_{S(0,1)} x^2_1 dx\). Theorem 2. Let \(\Omega\) be a bounded open set of \(\mathbb{R}^n\) with Lipschitz boundary, \(n\geq 2\), \(x_0\in \Omega\) and \(\delta> 0\) such that \(S(x_0, \delta)\subset\Omega\), \(0\leq u_D\in H^{1,2}(\Omega)\), let \(u\in H^{1,2}(\Omega)\) be the point of minimum of \(E:\{v\in H^{1,2}(\Omega), v-u_D\in H^{1,2}_0(\Omega)\}\to \mathbb{R}\), \(E(v)= \int_\Omega(|\nabla v|^2+ \max(v,0)) dx\), \(\Phi_{x_0}: ]0,\delta[\to \mathbb{R}\), \(\Phi_{x_0}(r)= r^{-n-2} \int_{S(x_0, r)}(|\nabla u|^2+ \max(u,0)) dx- 2r^{-n- 3} \int_{\partial S(x_0, r)} u^2 dH^{n- 1}\), then \(\Phi_{x_0}(\sigma)- \Phi_{x_0}(\rho)= \int^\sigma_\rho r^{-n- 2} \int_{\partial S(x_0, r)}2(\nabla u\cdot \nu- 2u/r)^2 dH^{n-1} dr\geq 0\) if \(0< \rho<\sigma<\delta\). Moreover, let \(x_0\in \Omega\cap \partial\{x\in \Omega: u(x)>0\}\), suppose that the inequality of Theorem 1 holds with \(\kappa\in ]0,1[\) for each \(c_r\), where \(c_r(x)= {|x|^2\over r^2} u\left(x_0+ {r\over|x|} x\right)\) such that \(r\leq r_0< 1\) and let \(u_0\) denote an arbitrary blow-up limit of \(u\) at \(x_0\). Then \(|\Phi_{x_0}(r)- \Phi_{x_0}(0+)|\leq |\Phi_{x_0}(r_0)- \Phi_{x_0}(0+)|\left({r\over r_0}\right)^{{(n+ 2)\kappa\over 1-\kappa}}\) for \(r\in ]0, r_0[\), there exists \(C(n, \kappa)\) such that \[ \int_{\partial S(0,1)} \Biggl|{u(x_0+ rx)\over r^2}- u_0(x)\Biggr|dH^{n-1}\leq C(n, \kappa)|\Phi_{x_0}(r_0)- \Phi_{x_0}(0+) |^{1/2}\Biggl({r\over r_0}\Biggr)^{{(n+ 2)\kappa\over 2(1- \kappa)}} \] for \(r\in ]0,r_0/2[\) and \(u_0\) is the unique blow-up limit of \(u\) at \(x_0\). Furthermore, the author obtains the differentiability of the solution at singular free boundary points and that the set of regular free boundary points is locally a \(C^{1,\beta}\) surface.
scientific article; zbMATH DE number 1355491