Regularity in an unusual variational problem
Given a symmetric positive definite \((2^n-1)\times (2^n-1)\) matrix \(Q\) the authors study an optimization problem of the following form: Find a function \(u:\mathbb R_+^n\to \mathbb R\) which minimizes the functional \[ J(u)=\int\int\dots\int_{R^n_+}[ (\partial_{x_1}\partial_{x_2}\dots \partial_{x_n}u)^2+(Q\overline D_{n-1}^nu,\overline D_{n-1}^nu)]\,dx_1\,dx_2\dots dx_n \] subject to \(u(0)=1\) where \(\overline D_{n-1}^nu=\{D^\alpha u| \alpha\in A_n,\,\alpha\notin(1,\dots,1)\}\) and \(A_n=\{\alpha=(\alpha_1,\dots,\alpha_n)\mid\) with \(\alpha_i=0\) or \(1\) for all \(i\}\). Let \(H_n(u)=\sum_{\alpha\in A}\| D^\alpha u\| ^2_{L^2}\) be a quadratic form and \(E_n\) be the Hilbert space consisted of functions \(u\) with \(H_n(u)<\infty\). The problem then is to find \[ \inf J(u), \quad u\in E_n,\, u(0)=1. \tag{1} \] The existence of a unique solution of (1) was proved a the previous paper by the authors. In this paper the authors prove that the solution of (1) is \(C^\infty\) up to and including the boundary.
scientific article; zbMATH DE number 5003841