# Systems of Semilinear Parabolic Variational Inequalities with Time-Dependent Convex Obstacles

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## Abstract

We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations.

## Keywords

Semilinear variational inequality Divergence form operator Time-dependent obstacle Penalization method Reflected backward stochastic differential equation## Mathematics Subject Classification

Primary 35K87 Secondary 60H30## 1 Introduction

*E*be a bounded domain in \({\mathbb {R}}^d\), \(E_T=[0,T]\times E\) and let \(D=\{D(t,x),(t,x)\in E_T\}\) be a family of uniformly bounded closed convex sets in \({\mathbb {R}}^m\) with nonempty interiors. In the paper we study the problem, which roughly speaking can be stated as follows: for measurable function \(\varphi :E\rightarrow {\mathbb {R}}^m\) such that \(\varphi (x)\in D(T,x)\), \(x\in E\), and measurable \(f:E_T\times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d}\rightarrow {\mathbb {R}}^m\) find \(u=(u^1,\dots ,u^m):E_T\rightarrow {\mathbb {R}}^m\) such that

*a*is defined and satisfies (1.4) in all \({\mathbb {R}}\times E\). In (1.3), \(\sigma \) is the symmetric square root of

*a*.

The main feature of the paper is that we deal with time-dependent obstacles. In case of single equation, i.e. when \(m=1\), problem (1.1), (1.2) is quite well investigated. For various results on existence, uniqueness and approximation of solutions in case of \(L^2\) data and one or two regular obstacles, i.e. when *D* has the form \(D(t,x)=\{y\in {\mathbb {R}}:\underline{h}(t,x)\le y\le {\bar{h}}(t,x)\}\) for some regular \(\underline{h} ,{\bar{h}}:E_T\rightarrow {\bar{{\mathbb {R}}}}\) (possibly \(\underline{h}\equiv -\infty \) or \({\bar{h}}\equiv +\infty \)) see the monograph [2, Sects. 2.2, 2.18] and more recent papers [4, 13, 15]. Linear problem of the form (1.1), (1.2) with \(L^2\) data and one irregular barrier is investigated in [20, 26]. For recent results on semilinear problem see [8] (one merely measurable obstacle) and [9] (two measurable obstacles satisfying some separation condition). The problem with two irregular obstacles and \(L^1\) data is investigated in [11].

In case of systems of equations the situation is quite different. To our knowledge, in this case only few partial results exist (see [20, Sect. 1.2] for the existence of solutions of weakly coupled systems and Example 9.3 and Theorem 9.2 in [18, Chap. 2] for the special case \(0\in D(t_2,\cdot )\subset D(t_1,\cdot )\) if \(0\le t_1\le t_2\); see also [14] for existence results concerning a different but related problem). The aim of the present paper is to prove quite general results on existence, uniqueness and approximation of solutions of (1.1), (1.2) in case the data are square integrable and *D* satisfies some mild regularity assumptions. The case of \(L^1\) data and irregular obstacles is more difficult but certainly deserves further investigation.

- (a)
\(u^i\) are quasi-continuous (with respect to the parabolic capacity determined by \(L_t\)) functions of class \(C([0,T];H)\cap L^2(0,T;H^1_0(E))\) and \(\mu ^i\) are smooth (with respect to the same capacity) measures of finite variation,

- (b)
*u*is a weak solution of the problemsuch that \(u(t,x)\in D(t,x)\) for quasi-every (q.e. for short) \((t,x)\in (0,T]\times E\),$$\begin{aligned} \frac{\partial u}{\partial t}+L_tu=-f_u-\mu ,\quad u(T, \cdot )=\varphi , \quad u(t, \cdot )_{|\partial E}=0,\,\,t \in (0,T), \end{aligned}$$ - (c)for every quasi-continuous function \(h=(h^1,\dots ,h^m)\) such that \(h(t,x)\in D(t,x)\) for q.e \((t,x)\in (0,T]\times E\),$$\begin{aligned} \sum ^m_{i=1}\int ^T_t\int _E(u^i-h^i)\,d\mu ^i\le 0,\quad t\in (0,T). \end{aligned}$$

*D*. Condition (c) is some kind of minimality condition imposed on \(\mu \). In case \(m=1\) it reduces to the usual minimality condition saying that \(\mu =\mu ^{+}-\mu ^{-}\), where \(\mu ^+\) (resp. \(\mu ^{-}\)) is a positive measure acting only when

*u*is equal to the lower obstacle \(\underline{h}\) (resp. upper obstacle \({\bar{h}}\)). Also remark that an important requirement in our definition is that

*u*is quasi-continuous and \(\mu \) is smooth. It not only ensures that the integral in (c) is meaningful, but also allows us to give a probabilistic representation of solutions. In fact, this probabilistic representation may serve as an equivalent definition of a solution of (1.1), (1.2).

*a*. As for

*D*, we assume that \((t,x)\mapsto D(t,x)\in \text{ Conv }\) is continuous if we equip \(\text{ Conv }\) with the Hausdorff metric. We also assume that

*D*satisfies the following separation condition: one can find a solution \(u_{*}\in {\mathcal {W}}\) of the Cauchy problem

*u*and \(\mu \) may be approximated by the penalization method. Note that our separation condition is not optimal, because we assume that \(\varepsilon >0\) and that \(u_{*}\) is more regular than the solution

*u*itself [see condition (a)]. The condition is also stronger than known sufficient separation conditions in the one dimensional case (see [9]). Nevertheless, it is satisfied in many interesting situations.

As in [8, 9, 11], to prove our result we use probabilistic methods. In particular, we rely heavily on the results of our earlier paper [12] devoted to reflected backward stochastic differential equations with time-dependent obstacles and in proofs we use the methods of the theory of Markov processes and probabilistic potential theory. Also note that the first results on multidimensional reflected backward stochastic differential equations were proved in [7] in case *D* is a fixed convex domain. The results of [7] were generalized in [25] to equations with Wiener–Poisson filtration. For related results with some time-depending domains see the recent paper [22].

## 2 Preliminaries

For \(x\in {\mathbb {R}}^m\), \(z\in {\mathbb {R}}^{m\times d}\) we set \(|x|^2=\sum ^m_{i=1}|x_i|^2\), \(\Vert z\Vert ^2=\text{ trace }(z^*z)\). By \(\langle \cdot ,\cdot \rangle \) we denote the usual scalar product in \({\mathbb {R}}^m\). Given a Hilbert space \({\mathcal {H}}\) we denote by \([{\mathcal {H}}]^m\) its product equipped with the usual inner product \((u,v)_{[{\mathcal {H}}]^m}=\sum ^m_{i=1}(u^i,v^i)_{{\mathcal {H}}}\) and norm \(\Vert u\Vert _{[{\mathcal {H}}]^m}=((u,u)_{[{\mathcal {H}}]^m})^{1/2}\). We will identify the space \([L^2(0,T;{\mathcal {H}})]^m\) with \(L^2(0,T;[{\mathcal {H}}]^m)\).

Throughout the paper, *E* is a nonempty bounded connected open subset of \({\mathbb {R}}^d\). \({\bar{E}}\) is the closure of *E* in \({\mathbb {R}}^d\), \(E^1={\mathbb {R}}\times E\), \(E_T=[0,T]\times E\), \(E_{0,T}=(0,T]\times E\). We set \(H=[L^2(E)]^m\), \(V=[H^1_0(E)]^m\), \(V'=[H^{-1}(E)]^m\), where \(H^{-1}(E)\) is the dual of the Sobolev space \(H^1_0(E)\). For \(u\in V\) we set \(|||\nabla u|||^2_H=\sum ^d_{k=1}\sum ^m_{i=1}\Vert \frac{\partial u^i}{\partial x_k}\Vert ^2_{L^2(E)}\).

The Lebesgue measure on \({\mathbb {R}}^d\) will be denoted by *m*. By \(m_1\) we denote the Lebesgue measure on \(E^1\).

### 2.1 Convex Sets and Functions

### 2.2 Time-Dependent Dirichlet Forms

In the paper by \(\text{ cap }\) we denote the parabolic capacity determined by the form \({\mathcal {E}}\) (for the construction and properties of \(\text{ cap }\) see [23, Sect. 4] or [24, Sect. 6.2]). We will say that some property is satisfied quasi-everywhere (q.e. for short) if it is satisfied except for some Borel subset of \(E^1\) of capacity \(\text{ cap }\) zero. Using \(\text{ cap }\) we define quasi-continuity as in [23]. By [23, Theorem 4.1] each function \(u\in {\mathcal {W}}\) has a quasi-continuous \(m_1\)-version, which we will denote by \({\tilde{u}}\).

Let \(\mu \) be a Borel signed measure on \(E^1\). In what follows \(|\mu |\) stands for the total variation of \(\mu \). By \({\mathcal {M}}_{0,b}(E^1)\) we denote the set of all Borel measures on \(E^1\) such that \(|\mu |\) does not charge sets of zero capacity \(\text{ cap }\) and \(|\mu |(E^1)<\infty \). By \({\mathcal {M}}_{0,b}(E_{0,T})\) we denote the subset of \({\mathcal {M}}_{0,b}(E^1)\) consisting of all measures with support in \(E_{0,T}\).

### 2.3 Markov Processes

*X*is the second component of \({\mathbf {X}}\), is a continuous time-inhomogeneous Markov process whose transition density \(p_E\) is the Green function for \(\frac{\partial }{\partial t}+L_t\) on \([0,T)\times E\) (for construction and properties of Green’s function see [1]).

Let \(\mu ,\nu \) be Borel measures on \(E^1\) and *E*, respectively. In what follows we write \(P_{\mu }(\cdot )=\int _{E^1}P_{s,x}(\cdot )\,d\mu (s,x)\), \(P_{s,\nu }(\cdot )=\int _{E}P_{s,x}(\cdot )\,\nu (dx)\). By \(E_{\mu }\) (resp. \(E_{s,\nu }\)) we denote the expectation with respect to \(P_{\mu }\) (resp. \(P_{s,\nu }\)).

Note that if *u* is quasi-continuous then it is \({\mathbf {M}}\)-quasi-continuous, i.e. for q.e. \((s,x)\in E^1\), \(P_{s,x}([0,\infty )\ni t\mapsto u({\mathbf {X}}_t)\) is continuous)=1 (see, e.g., [23, p. 298]).

*B*is \({\mathbf {M}}\)-exceptional, i.e.

## Remark 2.1

- (i)
From (2.6) and the fact that \(p_E(s,x,t,\cdot )\) is strictly positive on

*E*it follows that \(\text{ cap }(\{s\}\times \Gamma )>0\) for any \(s\in (0,T)\) and Borel set \(\Gamma \subset E\) such that \(m(\Gamma )>0\). Hence, if some property holds for q.e. \((s,x)\in (0,T)\times E\) then it holds for*m*-a.e. \(x\in E\) for every \(s\in (0,T)\). - (ii)
If \(P_{m_1}(\sigma _{B}<\infty )=0\) then in fact \(P_{s,x}(\sigma _{B}<\infty )=0\) for q.e. \((s,x)\in E^1\). This follows from the fact that the function \((s,x)\mapsto P_{s,x}(\sigma _{B}<\infty )\) is excessive (see remark at the end of [23, p. 294]).

*A*be a positive continuous additive functional of \({\mathbf {M}}\) and let \(\mu \in {\mathcal {M}}_{0,b}(E_{0,T})\) be a positive measure. We will say that

*A*corresponds to \(\mu \) (or \(\mu \) corresponds to

*A*) if for q.e. \((s,x)\in [0,T)\times E\),

*A*then \(\mu _1=\mu _2\). It is also known (see, e.g., Proposition (2.12) in [3, Chap. IV]) that if \(\mu \in {\mathcal {M}}_{0,b}(E_{0,T})\) corresponds to \(A^1\) and to \(A^2\) then \(A^1\) and \(A^2\) are equivalent, i.e. for each \(t\in [0,T]\), \(P_{s,x}(A^1_t=A^2_t)=1\) for q.e. \((s,x)\in [0,T)\times E\). If \(\mu \in {\mathcal {M}}_{0,b}(E_{0,T})\) then we say that \(\mu \) corresponds to

*A*if \(A=A^+-A^-\), where \(A^+,A^-\) are positive continuous additive functionals of \({\mathbf {M}}\) such that \(A^+\) corresponds to \(\mu ^+\) and \(A^-\) corresponds to \(\mu ^-\) (here \(\mu ^+\) (resp. \(\mu ^-\)) is the positive (resp. negative) part of the Jordan decomposition of \(\mu \)). Also note that (2.7) is some sort of the Revuz correspondence.

The following proposition is probably well known, but we do not have a reference.

## Proposition 2.2

## Proof

*u*satisfies (2.8), we see that the left-hand side of (2.12) equals \({\mathcal {E}}^{0,T}(u,v)\). From this and an analogue of [23, Theorem 7.4] for the form \({\mathcal {E}}^{0,T}\) it follows that \(N^{[u]}_t=-A_t\), \(t\in [0,T]\). Modifying slightly the proof of [23, (7.13)] we show (2.9). Finally, to show (2.10), let us denote by \(M^{u,k}\) the process on the right-hand side of (2.10) and consider a sequence \(\{u_n\}\) of smooth functions such that \(u_n\rightarrow u\) in \({\mathcal {W}}\). Then by the chain rule (see [29, Theorem 5.5]), \(M^{[u_n],k}=M^{u_n,k}\). It is clear that \(\{M^{u_n,k}\}_n\) is

*e*-convergent to \(M^{u,k}\). On the other hand, arguing as in the proof of [23, Theorem 7.2]) we show that \(\{M^{[u_n],k}\}_n\) is

*e*-convergent to \(M^{[u],k}\), which proves (2.10). \(\square \)

Let *Y* be a special \((({\mathcal {F}}_t),P_z)\)-semimartingale on [0, *T*], i.e. an \((({\mathcal {F}}_t),P_z)\)-semimartingale admitting a (unique) decomposition \(Y_t=Y_0+M_t+B_t\), \(t\in [0,T]\), with \(M_0=B_0=0\), *M* a local \((({\mathcal {F}}_t),P_z)\)-martingale and *B* an \(({\mathcal {F}}_t)\)-predictable finite variation process (see, e.g., [27, Sect. III.7]). Recall that the \({\mathcal {H}}^2(P_z)\) norm of *Y* is defined to be \(\Vert Y\Vert _{{\mathcal {H}}^2(P_z)}=(E_z|Y_0|^2)^{1/2}+(E_z[M]_T)^{1/2} +(E_z|B|_T^2)^{1/2}\), where [*M*] is the quadratic variation of *M* and \(|B|_T\) is the variation of *B* on the interval [0, *T*]. By \({\mathcal {H}}^2(P_z)\) we denote the space of all special \((({\mathcal {F}}_t),P_z)\)-semimartingales on [0, *T*] with finite \({\mathcal {H}}^2(P_z)\) norm.

## Remark 2.3

- (i)
Let \(z\in E_{0,T}\) and let \(\varphi ,f\) satisfy the assumptions of Proposition 2.2. Then \(M^{[u]}\) of Proposition 2.2 is a martingale under \(P_z\) (see [29, p. 327]), \(E_z|{\tilde{u}}({\mathbf {X}}_0)|^2=|{\tilde{u}}(z)|^2<\infty \), \(E_z[M^{[u],k}_{\cdot \wedge \zeta _{\tau }}]_T<\infty \), \(E_z(\int ^{\zeta _{\tau }}_0|f({\mathbf {X}}_t)|\,dt)^2<\infty \) for q.e. \(z\in E_{0,T}\). Hence \(Y={\tilde{u}}({\mathbf {X}}_{\cdot \wedge \zeta _{\tau }})\) is a semimartingale of class \({\mathcal {H}}^2(P_z)\) for q.e. \(z\in E_{0,T}\).

- (ii)Let \(M=(M^1,\dots ,M^d)\) be the AF of Proposition 2.2. Then
*M*is a martingale under \(P_z\) for q.e. \(z\in E_{0,T}\) (see [24, Sect. 5.1.2]). Setwhere \(\sigma ^{-1}\) is the inverse matrix of \(\sigma \). By Lévy’s theorem and (2.11), for q.e. \(z\in E_{0,T}\) the process \(B=(B^1,\dots ,B^d)\) is under \(P_z\) a$$\begin{aligned} B^i_t=\sum ^d_{j=1}\int ^t_0\sigma ^{-1}_{ij}({\mathbf {X}}_{\theta })\,dM^j_{\theta }, \quad t\ge 0,\quad i=1,\dots ,d, \end{aligned}$$(2.13)*d*-dimensional standard Brownian motion with respect to \(({\mathcal {F}}_t)_{t\ge 0}\). Finally, note that by (2.10) and (2.13),$$\begin{aligned} M^{[u],k}_t =\sum _{i,j=1}^d\int ^t_0\frac{\partial u^k}{\partial x_i}({\mathbf {X}}_{\theta })\sigma _{ij}({\mathbf {X}}_{\theta })\,dB^j_{\theta },\quad t\ge 0,\quad k=1,\dots ,m. \end{aligned}$$

## 3 Probabilistic Solutions of the Obstacle Problem

- (A1)
\(\varphi (x)\in D(T,x)\) for \(x\in E\), \(\varphi \in H\),

- (A2)
\(f(\cdot ,\cdot ,0,0)\in L^2(0,T;H)\),

- (A3)\(f:E_T\times {\mathbb {R}}^{m}\times {\mathbb {R}}^{m\times d}\rightarrow {\mathbb {R}}^m\) is a measurable function and there exist \(\alpha ,\beta \ge 0\) such thatfor all \(y_1,y_2\in {\mathbb {R}}^m\) and \(z_1,z_2\in {\mathbb {R}}^{m\times d}\).$$\begin{aligned} |f(t,x,y_1,z_1)-f(t,x,y_2,z_2)| \le \alpha |y_1-y_2|+\beta \Vert z_1-z_2\Vert \end{aligned}$$

*D*, we will need the following assumptions:

- (D1)
The sets

*D*(*t*,*x*) are bounded uniformly in \((t,x)\in E_T\) and the mapping \(E_T\ni (t,x)\mapsto D(t,x)\in \mathrm{Conv}\) is continuous. - (D2)
For some \(\varepsilon >0\) chosen so that Int\(D^{*}(t,x)\ne \emptyset \) for \((t,x)\in E_T\) and some \(f_{*}\in L^{2}(0,T;H)\) and \(\varphi _{*}\in H \) such that \(\varphi _{*}(x)\in D^{*}(T,x)\) for \(x\in E\) there exists a solution \(u_{*}\in {\mathcal {W}}\) of the Cauchy problem (1.5) such that \(u_{*}(t,x)\in D^{*}(t,x)\) for q.e. \((t,x)\in E_{0,T}\).

## Remark 3.1

- (D3)There exists a quasi-continuous function \(u_{*}:E_{0,T}\rightarrow {\mathbb {R}}^m\) such that the process \(Y^{*}\) defined as \(Y^{*}_t=u_{*}({\mathbf {X}}_{t\wedge \zeta _{\tau }})\), \(t\in [0,T]\), has the following properties:
- (a)
\(Y^{*}_t\in \text{ Int }D({\mathbf {X}}_{t\wedge \zeta _{\tau }})\) and \(\mathrm{dist}(Y^{*}_{t},\partial D({\mathbf {X}}_{t\wedge \zeta _{\tau }})) \ge \varepsilon \), \(t\in [0,T]\), \(P_{s,x}\)-a.s. for q.e. \((s,x)\in E_{0,T}\),

- (b)
\(Y^{*,i}\), \(i=1,\dots ,m\), is a special semimartingale of class \({\mathcal {H}}^2(P_{s,x})\) for q.e. \((s,x)\in [0,T)\times E\).

- (a)

*u*replaced by \(u_{*}\) and

*f*replaced by \(f_{*}\). Therefore \(Y^{*}\) satisfies condition (b) by remarks following Proposition 2 .3.

Condition (D2) is satisfied in the following natural situations.

## Example 3.2

Assume that one can find \(r>0\) such that for every \((t,x)\in E_T\), \(B(0,r)\subset D(t,x)\), where *B*(0, *r*) denotes the open ball with radius *r* and center at 0. Then *D* satisfies (D2). To see this, it suffices to consider the constant function \(u_{*}=(u_{*}^1,\dots ,u_{*}^m)=(0,\dots ,0)\). For instance, the above condition on *D* is satisfied if \(D(t,x)=\{y\in {\mathbb {R}}^m:\underline{h}^i(t,x)\le y_i\le {\bar{h}}^i(t,x),\,i=1,\dots ,m\}\), where \(\underline{h}^i,{\bar{h}}^i:[0,T]\times {\bar{E}}\rightarrow {\mathbb {R}}\) are continuous functions such that \(\underline{h}^i(t,x)<0<{\bar{h}}^i(t,x)\) for \((t,x)\in [0,T]\times {\bar{E}}\), \(i=1,\dots ,m\).

## Example 3.3

Let \(D(t,x)=\{y\in {\mathbb {R}}^m:\underline{h}^i(t,x)\le y_i\le {\bar{h}}^i(t,x),\,i=1,\dots ,m\}\), where \(\underline{h}^i,{\bar{h}}^i:[0,T]\times {\bar{E}}\rightarrow {\mathbb {R}}\) are continuous functions such that \(\underline{h}^i(t,x)<{\bar{h}}^i(t,x)\) for \((t,x)\in [0,T]\times {\bar{E}}\). Assume that there are continuous functions \(\underline{h}^i_0,{\bar{h}}^i_0:[0,T]\times {\bar{E}}\rightarrow {\mathbb {R}}\) such that \(\underline{h}^i(t,x)\le \underline{h}^i_0(t,x)<{\bar{h}}^i_0(t,x)\le {\bar{h}}^i(t,x)\) for \((t,x)\in [0,T]\times {\bar{E}}\), \(\underline{h}^i_0,{\bar{h}}^i_0\in {\mathcal {W}}_{0,T}\) and \((\frac{\partial }{\partial t}+L_t)\underline{h}^i_0,\, (\frac{\partial }{\partial t}+L_t){\bar{h}}^i_0\in L^2(0,T;L^2(E))\), \(i=1,\dots ,m\). Then (D2) is satisfied with \(u^i_{*}=({\bar{h}}^i_0+\underline{h}^i_0)/2\), \(i=1,\dots ,m\). Of course, the last condition for \(\underline{h}^i_0,{\bar{h}}^i_0\) is satisfied if \(\underline{h}^i_0,{\bar{h}}^i_0\in W^{1,2}_{2}((0,T)\times E)\) and \(\frac{\partial a_{ij}}{\partial x_k}\in L^{\infty }((0,T)\times E)\) for \(i,j,k=1,\dots ,d\).

In what follows we are going to show that under assumptions (A1)–(A3), (D1), (D2) there exists a unique solution of the problem (1.1), (1.2). It is convenient to start with probabilistic solutions. Solutions of (1.1), (1.2) in the sense of the definition given in Sect. 1 will be studied in the next section. Note that the definition formulated below is an extension, to the case of systems, of the probabilistic definition adopted in [9, 11] in case of single equation.

## Definition 3.4

- (a)
*u*is quasi-continuous, \(f^i_u\in L^1(E_{0,T})\), \(\mu ^i\in {\mathcal {M}}_{0,b}(E_{0,T})\), \(i=1,\dots ,m\), - (b)For q.e. \((s,x)\in E_{0,T}\) and \(i=1,\dots ,m\),where \(A^{\mu ^i}\), \(i=1,\dots ,m\), is the continuous additive functional of \({\mathbf {M}}\) associated with \(\mu ^i\) in the sense of (2.7),$$\begin{aligned} u^i({\mathbf {X}}_t)&=\varphi ^i({\mathbf {X}}_{\zeta _{\tau }}) +\int ^{\zeta _{\tau }}_{t\wedge \zeta _{\tau }}f^i_u({\mathbf {X}}_{\theta })\,d\theta +\int ^{\zeta _{\tau }}_{t\wedge \zeta _{\tau }}dA^{\mu ^i}_{\theta } \nonumber \\&\quad -\int ^{\zeta _{\tau }}_{t\wedge \zeta _{\tau }}\sigma \nabla u^i({\mathbf {X}}_{\theta })\,dB_{\theta },\quad t\in [0,T], \quad P_{s,x}\text{-a.s. }, \end{aligned}$$(3.1)
- (c)\(u(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\) and for every quasi-continuous function \(h=(h^1,\dots ,h^m)\) such that \(h(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\) we havefor q.e. \((s,x)\in E_T\).$$\begin{aligned} \int ^{\zeta _{\tau }}_0\langle u({\mathbf {X}}_t)-h({\mathbf {X}}_t),dA^{\mu }_t\rangle \le 0,\quad P_{s,x}\text{-a.s. } \end{aligned}$$(3.2)

## Remark 3.5

- (i)
From the fact that

*u*is quasi-continuous and \(u(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\) it follows that \(u({\mathbf {X}}_{t\wedge \zeta _{\tau }})\in D({\mathbf {X}}_{t\wedge \zeta _{\tau }})\), \(t\in [0,T]\), \(P_{s,x}\)-a.s. for q.e. \((s,x)\in E_{0,T}\). To see this, let us set \(B=\{(t,x)\in E_{0,T}:u(t,x)\not \in D(t,x)\}\) and \(\sigma _B=\inf \{t>0:{\mathbf {X}}_t\in B\}\). Since \(\text{ cap }(B)=0\), the set*B*is \({\mathbf {M}}\)-exceptional, and hence, by 2.1(ii), \(P_{s,x}(\sigma _B<\infty )=0\) for q.e. \((s,x)\in E_{0,T}\). Hence \(P_{s,x}({\mathbf {X}}_{t\wedge \zeta _{\tau }}\in B, t\in (0,T])=0\), which implies that \(u({\mathbf {X}}_{t\wedge \zeta _{\tau }})\in D({\mathbf {X}}_{t\wedge \zeta _{\tau }})\), \(t\in (0,T]\), \(P_{s,x}\)-a.s. for q.e. \((s,x)\in E_{0,T}\). In fact, we can replace (0,*T*] by [0,*T*], because \({\mathbf {X}}\) is right-continuous at \(t=0\) and*D*satisfies (D1). - (ii)
Conditions (3.1), (3.2) of the above definition say that under \(P_{s,x}\) the pair \((Y_t,Z_t)=(u({\mathbf {X}}_{t\wedge \zeta _{\tau }}),\sigma \nabla u({\mathbf {X}}_{t\wedge \zeta _{\tau }}))\), \(t\in [0,T]\), is a solution of the generalized Markov-type reflected BSDE with final condition \(\varphi \), coefficient

*f*, finite variation process \(A^{\mu }\) and obstacle*D*. - (iii)Taking \(t=0\) in (3.1) and then integrating with respect to \(P_{s,x}\) we see that for q.e. \((s,x)\in E_{0,T}\),By this and [10, Proposition 3.4], if \(u\in L^{2}(0,T;H)\) then \(u^i\in C([0,T];L^2(E))\), \(i=1,\dots ,m\).$$\begin{aligned} u^i(s,x)=E_{s,x}\Big (\varphi ^i({\mathbf {X}}_{\zeta _{\tau }}) +\int ^{\zeta _{\tau }}_0f^i_u({\mathbf {X}}_{\theta })\,d\theta +\int ^{\zeta _{\tau }}_0dA^{\mu ^i}_{\theta }\Big ),\quad i=1,\dots ,m. \end{aligned}$$
- (iv)
From continuity of \(A^{\mu ^i}\) and (2.7) one can deduce that \(\mu ^i(\{t\}\times E)=0\) for every \(t\in [0,T]\).

We begin with uniqueness of probabilistic solutions.

## Proposition 3.6

Assume (A3), (D1). Then there exists at most one probabilistic solution of OP\((\varphi ,f,D)\).

## Proof

## Proposition 3.7

*D*satisfies (D1) and there exists \(u_{*}=(u^1_{*},\dots u^m_{*})\in [{\mathcal {W}}]^m\) such that \(u_{*}\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\).

- (i)For every \(n\in {\mathbb {N}}\) there exists a unique strong solution \(u_n\in [{\mathcal {W}}]^m\) of the problem$$\begin{aligned} \frac{\partial u_n}{\partial t}+L_tu_n =-f_{u_n}+n(u_n-\Pi _{D(\cdot ,\cdot )}(u_n)),\quad u_n(T)=\varphi . \end{aligned}$$(3.4)
- (ii)There is
*C*depending only on \(\alpha ,\beta ,\Lambda \) and \(\Vert u_{*}\Vert _{[{\mathcal {W}}]^m}\) such that$$\begin{aligned} \sup _{0\le s\le T}\Vert u_n(s)\Vert ^2_H +\int ^T_0|||\nabla u_n(t)|||^2_H\,dt \le C(\Vert \varphi \Vert ^2_H+\Vert f(\cdot ,\cdot ,0,0)\Vert ^2_{L^{2}(0,T;H)}).\nonumber \\ \end{aligned}$$(3.5)

## Proof

*u*is a strong solution of (3.4) if and only if \({\hat{u}}=e^{\lambda t}u\) is a strong solution of (3.4) with \(L_t\) replaced by \(L_t-\lambda \), \(\varphi \) replaced by \(e^{\lambda T}\varphi \in H\) and

*f*replaced by some \({\hat{f}}\) still satisfying (A2) and (A3), without loss of generality we may replace \(L_t\) in (3.4) by \(L_t-\lambda \). Let \({\mathcal {A}}:L^2(0,T;V)\rightarrow L^2(0,T;V')\) be the operator defined as \(({\mathcal {A}}v)(t)= L_tv(t)-\lambda v(t) +f_{v(t)}-n(v(t)-\Pi _{D(\cdot ,\cdot )}(v(t)))\). By (D1) and (2.3) the mapping \((t,x,y)\mapsto -n(y-\Pi _{D(t,x)}(y))\) is continuous and Lipschitz continuous in

*y*for each fixed \((t,x)\in E_T\). Therefore for sufficiently large \(\lambda >0\) (depending on \(\alpha \) and \(\beta \)) the operator \({\mathcal {A}}\) is bounded, hemicontinuous, monotone and coercive, i.e. satisfies condition (7.84) from [18, Chap. 2]. Therefore the existence of a unique strong solution \(u_n\in [{\mathcal {W}}]^m\) of (3.4) follows from Theorem 7.1. and Remark 7.12 in [18, Chap. 2]. To prove (ii), set \(\varphi _{*}=u_{*}(T)\), \(f_{*}=-\frac{\partial u_{*}}{\partial t}-L_tu_{*}\). Then \(\varphi _{*}\in H\), \(f_{*}\in L^2(0,T;V')\), \(u_n-u_{*}\in [{\mathcal {W}}]^m\), \((u_n-u_{*})(T)=\varphi -\varphi ^{*}\) and

## Theorem 3.8

*D*satisfies (D1), (D2).

- (i)
There exists a unique solution \((u,\mu )\) of OP\((\varphi ,f,D)\).

- (ii)\(u^i\in C([0,T];H)\cap L^2(0,T;H^1_0(E))\), \(\mu ^i\in {\mathcal {M}}_{0,b}(E_{0,T})\), \(i=1,\dots ,m\), andwith constant$$\begin{aligned} \sup _{0<s\le T}\Vert u(s)\Vert ^2_H +\int ^T_0|||\nabla u(t)|||^2_H\,dt \le C(\Vert \varphi \Vert ^2_H+\Vert f(\cdot ,\cdot ,0,0)\Vert ^2_{L^{2}(0,T;H)})\nonumber \\ \end{aligned}$$(3.7)
*C*of Proposition 3.7. - (iii)Let \(u_n\in [{\mathcal {W}}]^m\) be a solution of (3.4). Thenand$$\begin{aligned} \Vert u_n-u\Vert ^2_{L^{2}(0,T;H)}\rightarrow 0, \qquad \int ^T_0|||\nabla (u_n-u)(t)|||^2_H\,dt \rightarrow 0 \end{aligned}$$(3.8)$$\begin{aligned} \mu _n\rightharpoonup \mu \quad \text{ weakly }\,{}^{*} \text{ on } (0,T]\times E. \end{aligned}$$(3.9)

## Proof

- (H1)
\(\xi \in D_T\), \(E_{s,x}|\xi |^2<\infty \),

- (H2)
\(E_{s,x}\int _0^T|{\bar{f}}(t,0,0)|^2\,dt<\infty \),

- (H3)\({\bar{f}}:[0,T]\times \Omega \times {\mathbb {R}}^m\times {\mathbb {R}}^{m\times d} \rightarrow {\mathbb {R}}^m\) is measurable with respect to \(Prog\otimes {\mathcal {B}}({\mathbb {R}}^m)\otimes {\mathcal {B}}^{m\times d}\) (
*Prog*denotes the \(\sigma \)-field of all progressive subsets of \([0,T]\times \Omega \)) and there are \(\alpha ,\beta \ge 0\) such that \(P_{s,x}\)-a.s. we havefor all \(y_1,y_2\in {\mathbb {R}}^m\) and \(z_1,z_2\in {\mathbb {R}}^{m\times d}\),$$\begin{aligned} |{\bar{f}}(t,y_1,z_1)-{\bar{f}}(t,y_2,z_2)| \le \alpha |y_1-y_2|+\beta \Vert z_1-z_2\Vert \end{aligned}$$ - (H4)
for each

*N*the mapping \(t\mapsto D_t\cap \{x\in {\mathbb {R}}^d:|x|\le N\}\in \text{ Conv }\) is càdlàg \(P_{s,x}\)-a.s. (with the convention that \(D_T=D_{T-}\)), and there is a semimartingale \(A\in {\mathcal {H}}^2(P_{s,x})\) such that \(A_t\in \text{ Int } D_t\), \(t\in [0,T]\), and \(\inf _{t\le T} \text{ dist }(A_t,\partial D_t)>0\).

*s*,

*x*) for which (3.10) is satisfied for all \(n\in {\mathbb {N}}\), i.e. for q.e. \((s,x)\in E_{0,T}\),

*Y*,

*K*of \(Y^{s,x},K^{s,x}\) not depending on

*s*,

*x*such that for q.e. \((s,x)\in E_{0,T}\),

*u*(still denoted by

*u*) such that

*u*is quasi-continuous and

*Y*is continuous, \([0,T]\ni t\mapsto u({\mathbf {X}}_{t\wedge \zeta _{\tau }})\) is continuous under \(P_{s,x}\) for q.e. \((s,x)\in E_{0,T}\), from which it follows that

*u*has a quasi-continuous \(m_1\)-version. Since \(u_n\rightarrow u\) \(m_1\)-a.e. and \(u_n(s,\cdot )\rightarrow u(s,\cdot )\)

*m*-a.e. for every \(s\in (0,T]\) [see Remark 2.1(i)], it follows from (3.5) that the first convergence in (3.8) holds true,

*n*), and a bounded measure \(\mu \) on \((0,T]\times E\) such that (3.9) is satisfied. By (2.7), for any \(f\in C_c((s,T)\times E\) and \(i=1,\dots ,m\) we have

*v*is strictly positive and continuous on \((s,T]\times E\). Let

*B*be a Borel subset of \((s,T]\times E\) such that \(\text{ cap }(B)=0\), and let

*H*be a compact subset of

*B*. By an approximation argument, (3.23) holds true for every bounded Borel function

*f*on \((s,T]\times E\). In particular, taking \(f={\mathbf {1}}_H\) in (3.23) we get

*H*of \(E_{0,T}\). This and [23, (4.4)]) imply that \(\mu (B)=0\). Thus \(\mu \) charges no set of capacity zero. Hence \(\mu \in {\mathcal {M}}_{0,b}(E_{0,T})\), which completes the proof. \(\square \)

It is perhaps worth remarking that in case the operator *L* is in nondivergence form [for instance, *L* is the Laplace operator \(\Delta \), or, more generally, \(\frac{\partial a_{ij}}{\partial x_k}\in L^{\infty }((0,T)\times E)\) for \(i,j,k=1,\dots ,d]\), then the process \({\mathbf {X}}\) corresponding to *L* can be constructed by solving an Itô equation. This allows simplifying some arguments in the proofs of the results presented above, but actually not much. One reason is that we are working in \(L^2\) setting, so even in the case where \(L=\Delta \) the solution of (1.1), (1.2) need not be continuous. On the other hand, since we use a stochastic approach via BSDEs [the basic relation is \(Y=u({\mathbf {X}})\), where *Y* is the first component of the solution of the corresponding BSDE; see Remark 3.5(ii)], we must know that *Y* is continuous, and hence that *u* is quasi-continuous. This in turn requires the introduction of quasi-notions (capacity, quasi-continuity, etc.) and we still have to use some results from the probabilistic potential theory.

## 4 Variational Solutions

In this section we show that results of Sect. 3 can be translated into results on solutions of (1.1), (1.2) in the sense of the analytical definition formulated below. Solutions in the sense of this definition will be called variational solutions.

## Definition 4.1

- (a)
*u*is quasi-continuous, \(u^i\in C([0,T];H)\cap L^2(0,T;H^1_0(E))\), \(\mu ^i\in {\mathcal {M}}_{0,b}(E_{0,T})\), \(i=1,\dots ,m\), - (b)for every \(\eta =(\eta ^1,\dots ,\eta ^m)\) such that \(\eta ^i\) is bounded and \(\eta ^i\in {\mathcal {W}}_0\) for \(i=1,\dots ,m\) we havefor all \(t\in [0,T]\), where \(\{a^{(t)}(\cdot ,\cdot ), t\ge 0\}\) is the family of bilinear forms on \(V\times V\) defined as$$\begin{aligned}&(u(t),\eta (t))_H+\int ^T_t\big \langle \frac{\partial \eta }{\partial s}(s),u(s)\big \rangle _{V',V}\,ds +\int ^T_ta^{(s)}(u(s),\eta (s))\,ds\nonumber \\&\qquad =(\varphi ,\eta (T))_H +\int ^T_t(f_u(s),\eta (s))_H\,ds +\int ^T_t\int _E\langle \eta ,d\mu \rangle \end{aligned}$$(4.1)$$\begin{aligned} a^{(t)}(\varphi ,\psi )=\frac{1}{2}\sum ^m_{k=1}\sum ^{d}_{i,j=1}\int _Ea_{ij}(t,x) \frac{\partial \varphi ^k}{\partial x_i}\frac{\partial \psi ^k}{\partial x_j}\,dx. \end{aligned}$$
- (c)\(u(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\), and for every quasi-continuous function \(h=(h^1,\dots ,h^m)\) such that \(h(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\),for all \(t\in (0,T)\).$$\begin{aligned} \int ^T_t\int _E\langle u-h,d\mu \rangle \le 0 \end{aligned}$$(4.2)

## Remark 4.2

## Remark 4.3

Assume that \(m=1\) and \(D(t,x)=\{y\in {\mathbb {R}}:\underline{h}(t,x)\le y\le {\bar{h}}(t,x)\}\) for some quasi-continuous \(\underline{h},{\bar{h}}:E_T\rightarrow {\mathbb {R}}\) such that \(\underline{h}<{\bar{h}}\). Then (4.2) reduces to the condition \(\int _{E_{0,T}} (u-\underline{h})\,d\mu ^+=\int _{E_{0,T}}({\bar{h}}-u)\,d\mu ^-=0\), where \(\mu ^+\) (resp. \(\mu ^-\)) is the positive (resp. negative) part of the Jordan decomposition of \(\mu \). The case of merely measurable obstacles is discussed in [8, 9, 11].

## Proposition 4.4

Assume (A3), (D1). Then there exists at most one variational solution of OP\((\varphi ,f,D)\).

## Proof

*u*is quasi-continuous, \([0,T]\ni t\mapsto u(\hat{\mathbf {X}}_t)\) is continuous \(P_{s,x}\)-a.s. for q.e. \((s,x)\in E_{0,T}\). Hence \(\alpha \int ^{\hat{\zeta }\wedge \tau (0)}_0e^{-\alpha t} u(\hat{\mathbf {X}}_t)\,dt=\int _0^{\alpha (\hat{\zeta }\wedge \tau (0))} e^{-t}u(\hat{\mathbf {X}}_{t/\alpha })\,dt \rightarrow u({\mathbf {X}}_0)\) \(P_{s,x}\)-a.s. as \(\alpha \rightarrow \infty \) for q.e. \((t,x)\in E_{0,T}\). Since \(u(t,y)\in D(t,y)\) for q.e. \((t,y)\in E_{0,T}\) and the sets

*D*(

*t*,

*y*) are uniformly bounded, it follows from this and the Lebesgue dominated convergence theorem that \(u_{\alpha }(s,x)\rightarrow E_{s,x}u({\mathbf {X}}_0)=u(s,x)\) for q.e. \((s,x)\in E_{0,T}\). Hence, by the Lebesgue dominated convergence theorem again,

## Theorem 4.5

Under the assumptions of Theorem 3.8 there exists a unique variational solution of OP\((\varphi ,f,D)\).

## Proof

*h*such that \(h(t,x)\in D(t,x)\) for q.e. \((t,x)\in E_{0,T}\) we have

*x*and using Remark 2.1(i) we obtain

## Remark 4.6

*u*of the solution of OP\((\varphi ,f,D)\) satisfies the following parabolic variational inequality: for every \(v\in {\mathcal {W}}\) such that \(v(t,x)\in D(t,x)\) for a.e. \((t,x)\in E_T\),

*E*is bounded. Therefore our results do not apply to single equations with one obstacle or, for instance, to systems of equations with obstacles of the form

*E*need not be bounded [see condition (H4) formulated in the proof of Theorem 3.8]. Another natural problem is to generalize the results of the present paper to systems involving more general operators, for instance nonsymmetric operators of the form \(L^b_t=L_t+\sum ^d_{i=1}b_i(x)\frac{\partial }{\partial x_i}\) with bounded measurable \(b:[0,T]\times E\rightarrow {\mathbb {R}}^d\). It is worth noting here that in [20, Sect. 1.2] the existence of solutions of variational inequalities with obstacles of the form (4.12) and operator \(L^b_t\) is proved in the special case where \(f=(f^1,\dots ,f^m)\) and \(f^i(t,x,y)=\sum ^m_{j=1}c_{ij}(t,x)y_j\) with bounded measurable \(c_{ij}\) such that \(c_{ij}\ge 0\) a.e. for \(i\ne j\) (in fact, in [20], the principal part and the first order part of the operator need not be the same in each equation of the system). Still another problem of interest is to generalize the results of the paper to irregular obstacles and/or \(L^1\) data (for one dimensional results in this direction see [8, 9, 11]).

## Notes

### Acknowledgments

This research was supported by NCN Grant No. 2012/07/B/ST1/03508.

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