Regularity in an unusual variational problem
type: Item
instance of: scholarly article
title: Regularity in an unusual variational problem
zbMATH Open document ID: 1113.49041
author: Ivar Ekeland
author: Louis Nirenberg
published in: Journal of Mathematical Fluid Mechanics
publication date: 2006-02-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 49K20
Mathematics Subject Classification ID: 91B28
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 5003841
zbMATH Keywords: Regularity
zbMATH Keywords: variational problems
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: mathematical finance
reviewed by: Yana Belopolskaya
MaRDI profile type: Publication
full work available at URL: s00021-005-0163-9
OpenAlex ID: W2049626071
DOI: 10.1007/S00021-005-0163-9
description: scientific article; zbMATH DE number 5003841