Backward stochastic differential equations and Dirichlet problems of semilinear elliptic operators with singular coefficients
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},\frac{\partial\phi}{\partial x_i}u\rangle, \langle q,u\phi\rangle,\end{aligned}\tag{\(2_b\)} \] where \(\langle\cdot,\cdot\rangle\) denotes the inner product in \(L^2(\mathbb{R}^d)\). From the introduction of the paper it is written: ``In this paper we will use probabilistic methods to solve the Dirichlet boundary value problem of the following form: \[ \begin{cases} \mathcal{A}u(x)=-f(x, u(x)),\quad \text{ for all } x\in D,\\ u(x)\mid D=\varphi,\quad \text{ for all } x\in\partial D,\end{cases}\tag{\(3_a\)} \] where \(\varphi\in C(\partial D)\), and \(f(x,y)\) is a nonlinear measurable function such that \[ \begin{cases} f\text{ is continuous for }y\in D,\\ (y_1-y_2)(f(x,y_1)- f(x,y_2))\leq -J(x)|y_1- y_2|^2,\\ |f(x,y)|\leq C(1+ _1(x)|)|y|\end{cases}\tag{\(3_b\)} \] for some constant \(C\) and \(J,J_1\in L^p(D)\)''. It is always assumed in the paper that there is a constant \(0<\lambda\leq 1\) such that \[ \begin{cases} \lambda|\xi|^2\leq \sum_{1\leq i\leq j\leq d} a_{ij}(x)\xi_i\xi_j,\quad \text{ for all }\xi\in \mathbb{R}^d, \text{ for all } x\in \mathbb{R}^d,\\ a_{ij}(x)\leq \frac{1}{\lambda}\quad \text{ for all } x\in \mathbb{R}^d,\;1\leq i\leq j\leq d.\end{cases}\tag{4} \] Furthermore, the authors assume in the main result of the paper (see Theorem 4.2) that each coefficient \(a_{ij}\) \((1\leq i\leq j\leq d)\) belongs to the VMO space, i.e., \[ \lim_{\rho\to 0}\,\sup_{r\leq\rho}\, \sup_{B_r\subset D}\,\frac{1}{ |B_r|} \int_{B_r} |a_{ij}(x)- a_{ij,B+r}|\,dx= 0,\tag{5} \] where \(a_{ij,B_r}= \frac{1}{|B_r|} \int_{B_r} |a_{ij}(x)\,dx\). Definition 1: The function \(u(x)\in W^{1,2}(D)\cap C(\overline D)\) is called a weak solution of problem (\(3_a\)) if \(u\) satisfies \[ \begin{cases} \mathcal{A}(u,\phi)= \int f(x,u(x))\phi(x)\,dx,\quad \text{ for all }\phi\in C^\infty_0(D),\\ u(x)=\varphi(x),\quad \text{ for all } x\in\partial D.\end{cases}\tag{6} \] The aim of the work under review is to show the existence of a weak solution for the Dirichlet problem (\(3_a\)), this paper may be considered as a continuation of previous ones, given by the first author and others for the linear case, i.e., \(f:=0\) (see [\textit{C.-Z. Chen} et al., J. Differ. Equations 260, No. 1, 26--55 (2016; Zbl 1333.35012)]). The main ideas of the paper are presented in the abstract as follows: ``In this paper, we prove that there exists a unique solution to the boundary Dirichlet boundary value problem for a general class of semilinear second-order elliptic differential operators which does not necessarily have a maximum principle and are non-symmetric in general. Our method is probabilistic. It turns out that we need to solve a class of backward stochastic differential equations (BSDEs) with singular coefficients, which is of independent interest itself. The theory of Dirichlet forms also play an important role''. To deal with the class of BSDEs with singular coefficients the authors justify their long technical stochastic computation given in Section 3 (approx. 10 pages!) as follows: ``It turns out that the classical \(L^2\) setting of BDSEs is not suitable here. We have to work in the framework of \(L^1\) and deal with class \(D\) stochastic processes,'' and refer the readers to [\textit{Ph. Briand} et al., Stochastic Processes Appl. 108, No. 1, 109--129 (2003; Zbl 1075.65503)] for more details. Assumption 1: (a) \(D\subset \mathbb{R}^d\) is a bounded, connected \(C^1\)-domain, (b) each \(a_{ij}\) \((1\leq i\leq j\leq d)\) is continuous and belongs to the VMO space (see (5)). The main result of the paper is given in Theorem 4.2 of the last Section 4. The proof of the mentioned theorem is based on the Assumption 1 plus some additional stochastic conditions denoted by (4.17) in the paper. The details are too complicated to expose in this review. We summarize the content of Theorem 4.2 in the Theorem 2 below. Theorem 2: Let the above quoted additional hypotheses be satisfied. Then there exists a unique continuous weak solution to the Dirichlet problem (\(3_a\)). Comments of the reviewer: 1) The authors make use of sophisticated probabilistic methods to solve the Dirichlet boundary value problem for elliptic p.d.o. with singular coefficients because they do not make use of the machinery of the theory of tempered distributions in the literature of functional analysis. The coefficients of the operator \(\mathcal{A}\) (see (\(1_a\))) are eventually some kind of ``distributions'' and so, one could not define rigorously the symbols writing in (\(2_a\)). The reviewer wonders that, without clarify the meaning of the symbol ``\(\operatorname{div}(\hat{b}u)\)'' how could the authors get the following duality formula in order to define the bilinear form \(\mathcal{Q}(u,\phi)\) (see (\(2_b\))), \[ \mathcal{Q}(u,\phi)=(-\mathcal{A}u,\phi)_{L^2}= (u,\mathcal{A}^*\phi)_{L^2},\; \text{ for all } u,\,\phi\in D(\mathcal{Q}). \] 2) The authors assume that each \(a_{ij}\) \((1\leq i\leq j\leq d)\) belongs to VMO functions in \(D\), and so it turns out that each \(a_{ij}\) is identified with the restriction in \(D\) of a BMO function. As consequence, each \(a_{ij}\) \((1\leq i\leq j\leq d)\) belongs to \(\mathcal{S}_d'(D)\), where \(\mathcal{S}_d'(D)\) denotes the space of tempered distributions in \(\mathbb{R}^d\) with support contained in \(\overline D\). Furthermore, since the authors assume that \(|b|^2 I_D\in L^p(D,dx)\) and \(|\hat{b}|^2 I_D\in L^p(D,dx)\) with \(p>\frac{d}{2}\) each coefficient \(b_i\), \(\hat{b_i}\) \((1\leq i\leq d)\) turns out to belong also to \(\mathcal{S}_d'(D)\). On the other hand, it is also natural to suppose that there exists a real \(N>0\) such that \((1+|x|^2)^{-N} f(x, u(x))\in L^p(D, dx)\) in order to conclude that \(f(x,u(x))I_D(x)\in\mathcal{S}_d'(D)\). Conclusion: The reviewer thinks that all the questions concerning the coefficients of the differential operator \(\mathcal{A}\) (see (\(1_a\))) raised in 1) above could have positive answers from the functional analytic point of view. Of course, in the Definition 1 of a weak solution it is natural to use the space of rapidly decreasing functions in \(\mathbb{R}^d\) with support in \(D\) as the space of test functions.
semilinear second order elliptic differential equations
backward stochastic differential equations
scientific article; zbMATH DE number 6920923
scientific article