Backward stochastic differential equations and Dirichlet problems of semilinear elliptic operators with singular coefficients
type: Item
instance of: scholarly article
title: Backward stochastic differential equations and Dirichlet problems of semilinear elliptic operators with singular coefficients
zbMATH Open document ID: 1425.60061
author: Saisai Yang
author: Tu-Sheng Zhang
published in: Potential Analysis
publication date: 2018-08-20T00:00:00Z
review text: By a semilinear second-order elliptic differential operator in a connected domain \(D\subseteq \mathbb{R}^p\) the authors always mean a partial differential operator (PDO) of the form, \[ \mathcal{A}u=\frac{1}{2} \sum_{1\leq i\leq j\leq d} \frac{\partial}{\partial x_i}\Biggl(a_{ij}\frac{\partial u}{\partial x_j}\Biggr)+ \sum_{1\leq i\leq d} b_i \frac{\partial u}{\partial x_i}-\text{``}\operatorname{div}(\hat{b_i} u)\text{''}+uq,\quad \text{ for all } u\in C^\infty_0(D),\tag{ \(1_a\)} \] in which, \[ \begin{aligned} &\text{a) } a_{ij}(x),\, b_j(x),\, \hat{b_j}(x)\text{ are a priori (Borel) measurable},\\ &\text{b) } C^\infty_0(D)\text{ is the space of smooth functions with compact support in }D.\end{aligned}\tag{\(1_b\)} \] It should be emphasized that the following terms in (\(1_a\)) \[ \frac{\partial u}{\partial x_i}\;\Biggl(a_{ij} \frac{\partial u}{\partial x_j}\Biggr)\;(1\leq i,j\leq d)\text{ and ``}\operatorname{div}(\hat{b}u)\text{''}\tag{\(2_a\)} \] are a priori just symbols and may not be differential operators. It is surprising that the reviewer, all-way through the paper, could not find the definition of the mysterious symbol ``\(\operatorname{div}(\hat{b}u)\)'', which is expected to be a linear first-order differential operator whose coefficients are nothing else than the ``distributional derivatives'' of \((a_{ij}\mid 1\leq i\leq j\leq d)\). The authors claim that the differential operator \(\mathcal{A}\) is characterized uniquely by its Dirichlet bilinear form \((\mathcal{Q},\mathcal{D}(\mathcal{Q}))\), in which \(\mathcal{D}(\mathcal{Q})\) is identified with the Sobolev space \(W^{1,2}(D)\). Namely, \[ \begin{aligned}\mathcal{Q}(u,\phi)&= (-\mathcal{A}u,\phi)_{L^2},\; \text{ for all } u,\,\phi\in D(\mathcal{Q})\\ &=\frac{1}{2} \sum_{1\leq i\leq j\leq d}\langle a_{ij},\frac{\partial u}{\partial x_j}\frac{\partial u}{\partial x_i}\rangle- \sum_{1\leq i\leq d}\langle b_i,\frac{\partial u}{\partial x_i}\phi\rangle- \sum_{1\leq i\leq d} \langle\hat{b_i},\fra…
reviewed by: Xuan Loc Nguyen
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 6920923
zbMATH Keywords: Dirichlet boundary value problem
zbMATH Keywords: semilinear second order elliptic differential equations
zbMATH Keywords: Dirichlet forms
zbMATH Keywords: backward stochastic differential equations
Wikidata QID: Q59527472
MaRDI profile type: Publication
full work available at URL: s11118-017-9654-6
OpenAlex ID: W2767753866
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DOI: 10.1007/S11118-017-9654-6
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description: scientific article; zbMATH DE number 6920923