UDC 517.9

MATHEMATICS

N. V. KRYLOV

A PROBLEM WITH TWO FREE BOUNDARIES FOR AN ELLIPTIC EQUATION AND OPTIMAL STOPPING OF A MARKOV PROCESS

(Presented by Academician A. N. Kolmogorov on 31 III 1970)

1. Let \(E_n\) be \(n\)-dimensional Euclidean space, \(U\) a bounded domain in \(E_n\) with twice continuously differentiable boundary,

\[ Lu(x)=a_{ij}(x)u_{ij}(x)+b_i(x)u_i(x)-c(x)u(x) \]

an elliptic differential operator acting on functions \(u\in W_p^2(U)\) (see \((^1)\)). Here \(u_{ij}\) is the mixed derivative with respect to \(x_i,x_j\); \(u_i\) is the derivative with respect to \(x_i\); summation over the indices \(i,j\) is assumed. Suppose that \(a_{ij}, b_i, c\) are bounded, \(c\ge 0\), the matrix \(a_{ij}\) is uniformly nondegenerate, and, for \(n\le 2\), \(p=2\), while for \(n>2\) the \(a_{ij}(x)\) are continuous in \(E_n\), \(p>n/2\). Let two functions \(\psi_1,\psi_2\in W_p^2(U)\) also be given such that \(\psi_1\ge \psi_2\) in \(U\), \(\psi_1=\psi_2\) on \(\partial U\).

Theorem 1. There exists, and moreover is unique, a function \(u\in W_p^2(U)\) possessing the following two properties: a) \(\psi_1(x)\ge u(x)\ge \psi_2(x)\) for \(x\in U\cup\partial U\); b) \((-1)^iLu\ge 0\) (a.e.) on

\[ \Gamma_i=\{x:\ (-1)^i[u(x)-\psi_i(x)]>0\} \]

for \(i=1,2\). In addition, if \(p>n\), then \(\operatorname{grad}(u-\psi_i)=0\) at all points of the set \(U\setminus\Gamma_i,\ i=1,2\).

The problem consisting in finding a function satisfying conditions a), b) is called a problem with two free boundaries, because when \(\psi_1>\psi_2\) in \(U\) it is equivalent to the problem of finding a function \(u\in W_p^2(U)\) and two closed sets \(G_1,G_2\) such that a) is satisfied, \(u=\psi_1\) and \(Lu\ge 0\) (a.e.) on \(G_1\), \(u=\psi_2\) and \(Lu\le 0\) (a.e.) on \(G_2\), and

\[ Lu=0 \]

(a.e.) on \(U\setminus(G_1\cup G_2)\).

This problem is a generalization of the problem with one free boundary and is directly related to problems with a variational inequality \((^2)\). Namely, if \(L\psi_1<0\), then from Theorem 1 it is not difficult to derive that \(u\) is the unique function in \(W_p^2(U)\) such that

\[ u\ge \psi_2,\qquad (u-\psi_2)|_{\partial U}=0, \]

\[ Lu\le 0 \]

(a.e.) on \(\{x:\ u(x)=\psi_2(x)\}\), and

\[ Lu=0 \]

(a.e.) on \(\{x:\ u(x)>\psi_2(x)\}\).

All these results can be obtained with the aid of certain facts from the theory of Markov processes. They are presented below.

2. Let \(X\) be a standard process in a locally compact space \(E\) (see \((^3)\)), \(B\) the space of Borel bounded functions \(f\) with norm

\[ |f|=\sup\{|f|:\ x\in E\}, \]

\(R_\lambda\) the resolvent of the process \(X\), \(m\) some probabilistic regular measure on \(E\), \(p>1\). Suppose that there exists a constant \(N\) such that for all \(x\in E\), \(f\in B\) the function \(R_0f(x)\) is continuous and

\[ R_0|f|(x)\le N\|f\|_p, \]

where \(\|f\|_p\) is the norm in the space \(L_p\) of functions integrable to the \(p\)-th power with respect to the measure \(m\). Let two Borel functions \(\psi_1(x),\psi_2(x)\) also be given, with \(\psi_1\ge\psi_2\). We want to study the function

\[ v(x)=\inf_{\tau}\sup_{\sigma} M_x\{\psi_1(x_\tau)\chi_{\tau<\sigma}+\psi_2(x_\sigma)\chi_{\sigma\le\tau}\}, \]

where the lower and upper bounds are taken over the set of all Markov times \(\tau\) and \(\sigma\), respectively.

The problem of studying \(v(x)\) is a generalization of the problems considered in \((^{4-6})\). Our method differs from the methods known from \((^{4-6})\); it consists in representing the function \(v(x)\) in the form \(R_0f(x)\), where \(f\in L_p\). In proving this representation we use the method of “continuous” stopping. The main technical tool is the following lemma.

For \(c\in B\) put
\[ R_c f(x)=M_x\int_0^\infty \exp\left[-\int_0^t c(x_s)\,ds\right]f(x_t)\,dt, \]
and, for \(n,m>0\), let
\[ u_{nm}=\inf\sup R_{c_1+c_2}(c_1\psi_1+c_2\psi_2), \]
where the \(\inf(\sup)\) is taken over the set of all functions \(c_1(c_2)\) such that \(0\le c_1\le n\) \((0\le c_2\le m)\),
\[ \delta_{nm}=-(u_{nm}-\psi_1)_+,\qquad \rho_{nm}=(u_{nm}-\psi_2)_-{}^* . \]

Lemma. Suppose that, for some \(f^i\in L_p\), \(\psi_i=R_0 f^i\) \((i=1,2)\). Then:

a) \(u_{nm}=R_c(n\delta_{nm}+m\rho_{nm})\);

b) if \(n>m\), then
\[ |\delta_{nm}|\le R_n f_-^1,\qquad \rho_{nm}\le R_m nR_n f_-^1+R_m f_+^2; \]

c)
\[ u_{nm}(x)=M_x\left[\int_0^{\tau\wedge\sigma}(n\delta_{nm}+m\rho_{nm})(x_t)\,dt +u_{nm}(x_{\tau\wedge\sigma})\right] \]
\[ =\sup_\sigma M_x\left[ \int_0^{\tau\wedge\sigma} n\delta_{nm}(x_t)\,dt -\rho_{nm}(x_\sigma)\chi_{\sigma\le\tau} +\psi_2(x_\sigma)\chi_{\sigma\le\tau} +u_{nm}(x_\tau)\chi_{\tau<\sigma} \right] \]
\[ =\inf_\tau \sup_\sigma M_x\left[ \psi_1(x_\tau)\chi_{\tau<\sigma} +\psi_2(x_\sigma)\chi_{\sigma\le\tau} -\delta_{nm}(x_\tau)\chi_{\tau<\sigma} -\rho_{nm}(x_\sigma)\chi_{\sigma\le\tau} \right]. \]

Proof. Consider the operator
\[ F_{nm}v=R_{n+m}\bigl[(n+m)v-n(v-\psi_1)_+ + m(v-\psi_2)_-\bigr] \]
in the space \(B\). From the inequality
\[ \bigl|[(m+n)t_1-n(t_1-a)_+ + m(t_1-b)_-] -[(m+n)t_2-n(t_2-a)_+ + m(t_2-b)_-]\bigr| \le (n+m)|t_1-t_2| \]
it follows that
\[ |F_{nm}v_1-F_{nm}v_2|\le (n+m)R_{n+m}|v_1-v_2|. \]
Further, note that from the estimate \(|R_0 f|\le N\|f\|\) it follows that
\[ (n+m)\|R_{n+m}f\|\le \varepsilon_{n+m}\|f\|, \]
where \(\varepsilon_{n+m}<1\) and does not depend on \(f\).

Thus the operator \(F_{nm}\) is a contraction in \(B\). By Banach’s theorem there exists \(v_{nm}\in B\) such that
\[ v_{nm}=F_{nm}v_{nm}. \]

Next, for \(0\le d\le k\) the equality
\[ R_d f=\sum_{i=0}^\infty [R_k(k-d)]^i R_k f \]
holds. Hence, for
\[ d=c_1+c_2,\qquad k=n+m,\qquad f=dv_{nm}-n(v_{nm}-\psi_1)_+ + m(v_{nm}-\psi_2)_-, \]
in view of the equality
\[ R_{n+m}f=c_{nm}+R_{n+m}(d-k)v_{nm}, \]
it easily follows that
\[ R_{c_1+c_2}f=v_{nm} \]
for \(0\le c_1\le n,\;0\le c_2\le m\).

From this equality and the inequality
\[ c_2(v_{nm}-\psi_2)+m(v_{nm}-\psi_2)_-\ge 0 \]
we obtain
\[ v_{nm}=\sup R_{c_1+c_2}\bigl[c_1v_{nm}-n(v_{nm}-\psi_1)_+ + c_2\psi_2\bigr], \]
and, as a consequence of the inequality
\[ c_1(v_{nm}-\psi_1)-n(v_{nm}-\psi_1)_+\le 0, \]
this gives
\[ v_{nm}=\inf\sup R_{c_1+c_2}(c_1\psi_1+c_2\psi_2), \]
i.e. \(v_{nm}=u_{nm}\). Assertion a) now follows from the equality
\[ R_{c_1+c_2}f=v_{nm} \]
for \(c_1=c_2=0\).

Let us again turn to the equality
\[ u_{nm}=R_{c_1+c_2}\bigl[(c_1+c_2)u_{nm}+n\delta_{nm}+m\rho_{nm}\bigr]. \]
For \(n>m,\;a\ge b\) we have
\[ nt-n(t-a)_+ + m(t-b)_-\le na, \]
therefore, for \(c_1=n,\;c_2=0,\;n>m\), we obtain
\[ u_{nm}\le nR_n\psi_1, \]
and since
\[ nR_n\psi_1=nR_nR_0 f^1=R_0 f^1-R_n f^1=\psi_1-R_n f^1, \]
it follows that
\[ u_{nm}\le \psi_1-R_n f^1\le \psi_1+R_n f_-^1 \]
and
\[ |\delta_{nm}|\le R_n f_-^1 . \]

If, however, we take \(c_1=0,\;c_2=m\), then
\[ u_{nm}\ge -R_m nR_n f_-^1+R_m[mu_{nm}+m\rho_{nm}] \ge -R_m nR_n f_-^1+mR_m\psi_2, \]
and
\[ \rho_{nm}\le R_m nR_n f_-^1+R_m f_+^2. \]
Assertion b) is proved.

The first equality in c) follows from a) and the strong Markov property of \(X\). The second equality is valid because, on the one hand,
\[ u_{nm}\ge \psi_2-\rho_{nm} \]
and \(\rho_{nm}\ge 0\), while, on the other hand, if
\[ \sigma=\inf\{t:\rho_{nm}(x_t)>0\}, \]
then
\[ \int_0^{\tau\wedge\sigma}\rho_{nm}(x_t)\,dt=0 \]
and
\[ \psi_2(x_\sigma)-\rho_{nm}(x_\sigma)=u_{nm}(x_\sigma). \]
The third equality is obtained by analogous arguments from the second. The lemma is proved.

From this lemma the following theorems are derived rather simply. In their statements,
\[ M_x(\tau,\sigma)=M_x[\psi_1(x_\tau)\chi_{\tau<\sigma}+\psi_2(x_\sigma)\chi_{\sigma\le\tau}]. \]

Theorem 2. Suppose that, for some \(f^i\in B\), \(\psi_i=R_2 f^i\) \((i=1,2)\); then there exists a function \(f\in B\) such that \(v=R_0 f\), and hence \(v\) is continuous. Further,
\[ (-1)^i f\le 0\quad\text{on}\quad \Gamma_i(v)=\{x:(-1)^i[v(x)-\psi_i(x)]>0\}, \]
\[ \lim_{n\to\infty}\lim_{m\to\infty}\|v-u_{nm}\|=0,\qquad v(x)=\sup_\sigma M_x(\tau_1,\sigma)=\inf_\tau M_x(\tau,\tau_2), \]
\[ {}^*\quad a_+=\frac12(|a|+a),\qquad a_-=\frac12(|a|-a). \]

where \(\tau_i=\inf\{t:\ v(x_t)=\psi_i(x_t)\}\) \((i=1,2)\). Moreover, if \(u,h\in B\) are functions related by the relation \(u=R_0h\), \(\psi_1\geq u\geq\psi_2\), and \((-1)^i h\leq 0\) on \(\Gamma_i(u)\), for \(i=1,2\), then \(u=v\).

Theorem 3. Suppose there exists a constant \(N_1\) such that \(\lambda\|R_\lambda f\|_p\leq N_1\|f\|_p\) for all \(f\in B\), \(\lambda>0\). Then in the formulation of Theorem 2 one may replace \(B\) by \(L_p\). Moreover, as \(\tau_i\) one may also take
\[ \inf\{t:\ v(x_t)=\psi_i(x_t),\quad x_t\in \operatorname{supp}[(-1)^i f]^+\}. \]

Before formulating Theorem 4, take two closed sets \(G_1,G_2\) and denote by \(\mathfrak M_i\) the set of Markov times such that
\[ P_x\{x_\tau\in G_i\}=P_x\{\tau<\zeta\} \]
for all \(x\), and by \(\mathfrak N_i\) the set of hitting times of closed subsets of \(G_i\).

Theorem 4. Suppose the assumption of Theorem 3 is satisfied and there exist sequences of functions \(f_n^i\in L_p\) such that
\[ \operatorname{supp}[(-1)^i f_n^i]^+\subset G_i,\qquad R_0 f_n^1\geq R_0 f_n^2,\qquad \lim_{n\to\infty}\|[\psi_i-R_0 f_n^i]\chi_{G_i}\|=0. \]
Then
\[ \inf_{\tau\in\mathfrak M_1}\times \sup_{\sigma\in\mathfrak M_2} M_x(\tau,\sigma) = \inf_{\tau\in\mathfrak N_1}\sup_{\sigma\in\mathfrak M_2} M_x(\tilde\tau,\sigma) = \inf_{\tau\in\mathfrak M_1}\sup_{\sigma\in\mathfrak N_2^{\,x}} M_x(\tau,\sigma). \]
Moreover, all expressions in the last equality are continuous.

We make some remarks concerning the proofs of Theorems 1–4. Theorem 1 is derived from Theorem 3 with the aid of the known properties of \(R_0 f\) for quasidiffusion processes; Theorem 4 is also easily derived from Theorem 3. The proofs of Theorems 2 and 3 follow one and the same plan. First, with the aid of Lemma b), it is proved that
\[ \lim_{n\to\infty}\lim_{m\to\infty}\|\rho_{nm}+|\delta_{nm}|\|=0; \]
then, using Lemma c), that \(\|u_{nm}-v\|\to 0\). After this, from the estimate \(R_0|f|\leq N\|f\|_p\) and Lemmas a), b), one obtains the equality \(v=R_0 f\) and the sign properties of \((-1)^i f\). The equality
\[ v(x)=\sup_{\sigma} M_x(\tau_1,\sigma) \]
follows from the equality of \(u_{nm}\) to the third term in equality c) of the lemma and from the estimates \(|\delta_{nm}|\leq R_n f^{-}\), \(R_0|f|\leq N\|f\|_p\), \(\lambda\|R_\lambda f\|_p\leq N_1\|f\|_p\). The last assertion of Theorem 2 is almost obvious.

Moscow State University
named after M. V. Lomonosov

Received
19 III 1970

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