A construction of a uniform continuous minimizing movement associated with a singular functional (Q2360064)

This interesting paper is devoted to a minimization problem concerning a singular functional that appears in a free boundary problem. The author discusses a method of constructing a uniformly continuous minimizing movement (MM) for the singular functional introduced by \textit{H. W. Alt} and \textit{L. A. Caffarelli} [J. Reine Angew. Math. 325, 105--144 (1981; Zbl 0449.35105)], and denoted by AC for short. A metric space \(X\) with metric \(d\) and a function \(F: X\to [0,\infty ]\), \(F(u_0)< +\infty\), are considered, where the initial data \(u_0\in X\). For some \(h>0\), \(u_{0,h}=u_0\), and recursively define \(u_{n,h}\) (\(n\in \mathbb{N}\)) as the minimizer of the variational problem: \(\text{Minimize} \;d(u,u_{n-1,h})^2/h+F(u)\) among all \(u\in X\). Thus define the sequence \((u_{n,h})_{n=0}^{\infty }\). Then the discrete minimizing movement (DMM(\(F;u_0,h\)) for short) is the collection of all such sequences. We say that \(u:[0,\infty )\to X\) is a MM for \(F\) starting from \(u_0\) if there exists a sequence \((h_j)_{j=1}^{\infty }\) converging to zero as \(j\to\infty \) and \((u_{n,h})_{n=0}^{\infty }\in \)DMM(\(F;u_0,h_j\)) such that \(\lim\limits_{j\to\infty }u_{h_j}(t)=u\) in \(X\) (\(t\geq 0\)); the colection of all functions \(u(t)\) is called the MM of \(F\) starting from \(u_0\), and designated by MM(\(F;u_0\)). In the problem under consideration, the author assumes that the initial function belongs to the Sobolev class \(W^{1,2}(\mathbb{R}^N) \cap W^{2,\infty }(\mathbb{R}^N)\cap C^{1,\alpha }(\mathbb{R}^N)\) for \(\alpha\in (0,1)\), and is bounded in addition. The following functional is defined: \[ \text{AC}(u)\equiv \begin{cases} \int\limits_{\mathbb{R}^N} (|\nabla u|^2+\chi (u)) \;\;& \text{for} \;\;u\in W^{1,2}(\mathbb{R}^N), \\ +\infty \;\;\;\;\;\;\;& \text{for} \;\;u\in L^2(\mathbb{R}^N)\setminus W^{1,2}(\mathbb{R}^N), \end{cases} \] where \(\chi (\cdot )\) is the characteristic function of the interval \((0,\infty )\). Then there exists a function \(u=u(t)\in \text{MM(AC;}u_0)\) such that \(0\leq u\leq \sup\limits_{\mathbb{R}^N}u_0\) in \([0,\infty )\times\mathbb{R}^N\), and the following estimates hold: \[ |u(t,x)-u(t,x')|\leq C_1|x-x'|, \quad t\geq 0, \;x,x'\in\mathbb{R}^N, \] and \[ |u(t,x)-u(t',x)|\leq C_2|t-t'|^{1/2},\quad t,t'\geq 0,\; x\in\mathbb{R}^N. \] The constants \(C_1,C_2\) are positive, the first depends on \(u_0\), and the second on both \(u_0\) and \(N\).