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A homogeneity improvement approach to the obstacle problem. (English) Zbl 0940.35102
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.
Reviewer: G.Bottaro (Genova)

MSC:
 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 35B65 Smoothness and regularity of solutions to PDEs 49J10 Existence theories for free problems in two or more independent variables
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