UDC 917.946.9

ON THE FIRST BOUNDARY VALUE PROBLEM FOR ELLIPTIC EQUATIONS OF SECOND ORDER

N. V. KRYLOV

At the present time there is a large number of papers by various authors in which methods of the theory of Markov processes are applied to the study of properties of solutions of elliptic differential equations of second order *). The possibility of such an application is based on replacing the elliptic operator by the infinitesimal (or characteristic) operator of the corresponding Markov process.

In this case the solutions of the resulting (extended) equation are always written explicitly in the form of known expressions, and therefore, in order to study properties of solutions of differential equations, one may study the properties of these expressions. Of course, the most important questions here are those concerning the existence of derivatives and boundary values.

In the case when the coefficients of the equation satisfy the Hölder condition, the first question is solved with the aid of existence theorems for differential equations and uniqueness theorems for the solution of the extended equation; the second, with the aid of a general probabilistic criterion for the regularity of a point and estimates of the fundamental solution of the corresponding parabolic equation (see [1]).

The purpose of this paper is to apply such an approach to the first boundary value problem in the case when weaker restrictions than the Hölder condition are imposed on the coefficients of the operator. We shall mainly deal with the solution of the two questions described above. It should be said that here one has to apply methods somewhat different from those in [1].

Let \(V\) be a bounded domain **) in Euclidean space \(E_l\) of dimension \(l\), \(\overline V\) the closure of \(V\), and \(\partial V\) the boundary of \(V\). Let \(B(\partial V)\) be the space of bounded measurable functions on \(\partial V\) with norm
\(\|u\|_{B(\partial V)}=\sup_{x\in \partial V}|u(x)|\), \(C(V)\) the space of continuous bounded functions on \(V\) with norm
\(\|u\|_{B(V)}=\sup_{x\in V}|u(x)|\), \(C_0(V)\) the subspace of the space \(C(V)\) consisting of functions continuous up to the boundary \(V\) and equal to zero on \(\partial V\); \(C^2(V)\) the space of twice continuously differentiable functions on \(V\), \(C_0^2(V)\) the subspace of \(C^2(V)\) consisting of functions equal to zero on \(\partial V\). Let \(p>1\) and let \(L_p(V)\) be the space of functions for which

\[ \|u\|_{L_p(V)}=\left(\int_V |u(x)|^p\,dx\right)^{1/p}<\infty; \]

\(W_p^2(V)\) is the closure of the set \(C^2(V)\) in the norm

*) See, for example, [1]; further references may also be found there.

**) All domains considered will be assumed bounded.

\[ \|u\|_{W_p^2(V)} = \left( \sum_{i,j=1}^{l}\left\|\frac{\partial^2 u}{\partial x_i \partial x_j}\right\|_{L_p(V)}^p + \|u\|_{L_p(V)}^p \right)^{1/p}, \]
\(\overset{\circ}{W}{}_p^2(V)\) denotes the closure of the set \(C_0^2(V)\) in the same norm.

Consider the elliptic operator

\[ L=L(x)= \sum_{i,j=1}^{l} a_{ij}(x)\frac{\partial^2}{\partial x_i\partial x_j} + \sum_{i=1}^{l} b_i(x)\frac{\partial}{\partial x_i} - c(x), \]

acting on functions \(u\in W_p^1(V)\). We shall assume that \(b_i(x)\), \(c(x)\) are measurable functions and

1) for arbitrary \(\lambda_i\) and \(x\in E_l\),

\[ \mu \sum_{i=1}^{l}\lambda_i^2 \le \sum_{i,j=1}^{l} a_{ij}(x)\lambda_i\lambda_j \le \mu^{-1}\sum_{i=1}^{l}\lambda_i^2, \]

where \(\mu=\mathrm{const}>0\);

2) if \(b(x)=(b_1(x),\ldots,b_l(x))\), then \(\|b(x)\|,\ |c(x)|\le M<\infty,\quad c(x)\ge 0\);

3) let

\[ w(r)=\sup_{i,j,x}\ \sup_{\|h\|\le r}|a_{ij}(x)-a_{ij}(x+h)|, \]

then \(w(r)\to 0\) as \(r\to 0\).

Under these conditions, in the works of H. Tanaka [2] and the author [3] a continuous Markov process \(X=(x_t,\zeta,M_t,P_x)^*)\) was constructed, corresponding to the operator \(L\) in the space \(E_l\). Here \(x_t=x_t(\omega)\) is the position of the trajectory of the process at time \(t\in[0,\zeta)\); \(\omega\) is a point of some space of elementary events \(\Omega\); \(\zeta\) is the random time of termination of the trajectory; \(M_t\) are \(\sigma\)-algebras of subsets of \(\Omega\); \(P_x\) are probability distributions on the space \(\Omega\). Let \(\tau\) be the time of first exit of a trajectory of the process \(X\) from the domain \(V\), and put

\[ R(V)f(x)=M_x\int_{0}^{\tau} f(x_t)\,dt^{**)}. \]

Then, by Theorem 4 of [3], for any twice continuously differentiable function \(u\) and point \(x\in V\) the equality

\[ u(x)=R(V)f(x)+M_x\varphi(x_\tau), \tag{1} \]

holds, where \(f(x)=-L(x)u(x)\) and \(\varphi(x)=u(x)\) for \(x\in\partial V\).

Now consider in the domain \(V\) the equation

\[ L(x)u(x)=-f(x)\quad \text{for } x\in V,\qquad u(x)=\varphi(x)\quad \text{for } x\in\partial V. \tag{2} \]

If there exists \(u\in C^2(V)\) such that the equalities (2) are satisfied, then \(u(x)\) has the form (1). Therefore, by formula (1), we obtain, so to speak, a generalized solution of (2) for any \(f\) and \(\varphi\) for which the right-hand side in (1) makes sense\(^{***)}\).

For what follows we note an important property of the function \(u(x)\) defined by formula (1).

* We use the notation and terminology from the book [1].

** If \(\xi(\omega)\) is a random variable, then \(M_x\xi\) is, by definition, equal to \(\displaystyle \int_{\Omega}\xi(\omega)P_x(d\omega)\).

*** (1) also gives the solution of the corresponding extension of (2), discussed above.

Lemma (Dynkin [1]). Let \(U\) be some domain, \(\tau_1\) the time of the first exit of \(X\) from \(U\cap V\), \(x\in U\cap V\), and suppose that at the point \(x\) the right-hand side in (1) has meaning; then

\[ u(x)=R(U\cap V)f(x)+M_x\overline{\varphi}(x_{\tau_1}), \]

where \(\overline{\varphi}(x)=\varphi(x)\) for \(x\in \partial V\), and \(\overline{\varphi}(x)=u(x)\) otherwise.

§ 1. Smoothness of the generalized solution inside the domain

In this section we proceed to the solution of the first of the questions posed. The basic result for us is

Theorem 1. Let \(p>l/2,\ p>1,\ f\in L_p(V), \varphi\in B(\partial V)\); then formula (1) defines a function \(u\) belonging to \(W_p^2(V')\) for every domain \(V'\) such that \(\overline{V'}\subset V\). Moreover, \(Lu=-f\).

The proof of this theorem is based on the following lemmas.

Lemma 1. There exist a function \(g(x,y)=g_V(x,y)\) and a constant \(C\) such that for every function \(f\in L_p(V)\) and point \(x\in V\)

\[ R(V)f(x)=\int_V g(x,y)f(y)\,dy \quad\text{and}\quad |R(V)f(x)|\le C\|f\|_{L_p(V)} . \]

Proof. It is clear that it suffices to consider only the case \(f\ge 0\). Let \(V_1\) be a domain such that \(V_1\supset \overline{V}\) and the boundary of \(V_1\) is twice continuously differentiable. Suppose that \(f\in C_0(V_1)\). By Corollary 2 of Theorem 3 [3], the function \(u=R(V_1)f\) is the unique solution of the equation

\[ Lu=-f \]

in \(\overset{\circ}{W}{}_p^2(V_1)\). According to the remark after Theorem 15.2 [4], it follows that there exists a constant \(C_1\) such that

\[ \|u\|_{W_p^2(V_1)}\le C_1\|Lu\|_{L_p(V_1)} \tag{3} \]

for all \(u\in \overset{\circ}{W}{}_p^2(V_1)\).

By the Sobolev embedding theorem [5], for some constant \(C_2\), \(|u(x)|\le C_2\|Lu\|_{L_p(V_1)}\) for all \(u\in \overset{\circ}{W}{}_p^2(V_1)\), \(x\in V_1\). Further, evidently, \(R(V)f(x)\le R(V_1)f(x)\), and hence \(|R(V)f(x)|\le C_2\|f\|_{L_p(V_1)}\). In this inequality the left-hand side does not depend on the values of \(f(x)\) outside \(V\), and, consequently, for every \(f\in C(V)\), \(|R(V)f(x)|\le C_2\|f\|_{L_p(V)}\).

The proof of Lemma 1 can now be completed in the same way as the proof of Lemma 6 [3].

Lemma 2. Suppose that \(\partial V\) is twice continuously differentiable; then for every function \(f\in L_p(V)\) there exists a unique solution of the problem

\[ Lu=-f \]

in \(\overset{\circ}{W}{}_p^2(V)\). This solution is given by the formula \(R(V)f(x)\).

Proof. The assertion of this lemma in the case \(f\in C_0(V)\) coincides with Corollary 2 of Theorem 3 [3].

For arbitrary \(f\in L_p(V)\), choose a sequence \(\{f_n;\ n\ge 1\}\subset C_0(V)\) such that \(\|f_n-f\|_{L_p(V)}\to 0\) as \(n\to\infty\). Then, by Lemma 1, \(R(V)f_n(x)\to R(V)f(x)\) as \(n\to\infty\) at each point \(x\in V\). Moreover, from estimate (3) we obtain that the sequence \(\{R(V)f_n;\ n\ge 1\}\) converges in \(\overset{\circ}{W}{}_p^2(V)\) to some function \(u(x)\). Evidently \(Lu=-f\). Finally, by …

by Sobolev’s embedding theorem [5] we obtain that \(R(V)f_n(x)\) converge in \(C(V)\) to \(u(x)\), and hence \(u(x)=R(V)f(x)\).

The lemma is proved.

We shall complete the preparation for the proof of Theorem 1 with Lemma 3.

Lemma 3. Suppose that the principal part of the operator \(L\) at the point \(0\) is the Laplace operator \(\Delta\). Denote by \(S(r)\) the ball of radius \(r\) with center at the point \(0\). Then for every sufficiently small \(r\) and every \(r_1\in(0,r)\) there exists a constant \(C\) such that, for all \(u\in W_p^2(S(r))\),

\[ \|u\|_{W_p^2(S(r_1))}\leq C\bigl(\|Lu\|_{L_p(S(r))}+\|u\|_{B(\partial S(r))}\bigr). \]

Proof. Obviously, it is enough to prove this lemma only for smooth \(u(x)\). Let \(u(x)\) be a smooth function, and let \(u_1(x)\) and \(u_2(x)\) be such that \(\Delta u_1=\Delta u\) and \(u_1(x)=0\) for \(x\in\partial S(r)\), while \(\Delta u_2=0\) and \(u_2(x)=u(x)\) for \(x\in\partial S(r)\). It is clear that \(u=u_1+u_2\).

From estimate (3), using similarity considerations, it is easy to obtain that there exists a constant \(C_1\) such that, for all \(r\leq 1\),

\[ \|u_1\|_{W_p^2(S(r'))}\leq \|u_1\|_{W_p^2(S(r))}\leq C_1\|\Delta u_1\|_{L_p(S(r))}. \]

Moreover, from the representation of \(u_2\) by means of the Poisson integral and similarity considerations, it is also not difficult to obtain that there is a constant \(C_2\) such that, for \(r\leq 1\) and \(r'\in(0,r)\),

\[ \|u_2\|_{W_p^2(S(r'))}\leq C_2(r-r')^{-l-2}\|u\|_{B(\partial S(r))}. \]

Thus, for any smooth function \(u\) and \(r\leq 1\), for all \(r'\in(0,r)\),

\[ \|u\|_{W_p^2(S(r'))}\leq C_3\bigl(\|\Delta u\|_{L_p(S(r))}+(r-r')^{-l-2}\|u\|_{B(\partial S(r))}\bigr), \tag{4} \]

where \(C_3=\max(C_1,C_2)\). Let us also note that representation (1) is valid for \(u\), and hence, by Lemma 1,

\[ |u(x)|\leq C_4\|Lu\|_{L_p(S(r))}+\|u\|_{B(\partial S(r))}. \tag{5} \]

Now let \(r\) be such that \(C_3\omega(2r)=\delta<\dfrac{1}{2}\) and \(r'\in(0,r)\). Choose \(\varepsilon\) so that

\[ \varepsilon\sum_{k=1}^{\infty} k^{-2}=r-r'. \]

Put

\[ S_i=S\left(r-\varepsilon\sum_{k=i}^{\infty} k^{-2}\right)\qquad (i=1,2,\ldots). \]

From (4), (5), and the obvious equality \(\Delta u=Lu+(\Delta-L)u\), for all \(k\geq 1\) we obtain

\[ \|u\|_{W_p^2(S_k)}\leq C_3\Bigl[(1+C_4\varepsilon^{-l-2}k^{2l+4})\|Lu\|_{L_p(S(r))}+ \]

\[ +\|(\Delta-L)u\|_{L_p(S_{k+1})}+\varepsilon^{-l-2}k^{2l+4}\|u\|_{B(\partial S(r))}\Bigr]. \tag{6} \]

Further, the second term in brackets does not exceed

\[ \delta\|u\|_{W_p^2(S_{k+1})} + C_5\sum_{i=1}^{l}\left\|\frac{\partial u}{\partial x_i}\right\|_{L_p(S_{k+1})}, \]

where \(C_5\) does not depend on \(u\) and \(k\). From the known inequa-

... (see [4], Lemma (4.1)) it follows that there exists a constant \(C_6\) such that

\[ C_5\sum_{i=1}^{l}\left\|\frac{\partial u}{\partial x_i}\right\|_{L_p(S_k)} \leq \frac12 \|u\|_{W_p^2(S_k)}+C_6\|u\|_{L_p(S(r))}. \]

Consequently, from (5) and (6), for \(k \geq 1\),

\[ \|u\|_{W_p^2(S_k)} \leq \delta_1\|u\|_{W_p^2(S_{k+1})} + C_7 k^{2l+4}\bigl(\|Lu\|_{L_p(S(r))}+\|u\|_{B(\partial S(r))}\bigr), \]

where \(C_7\) does not depend on \(k\) or \(u\), \(\delta_1=\dfrac12+\delta<1\). Multiplying this inequality by \(\delta_1^k\) and summing over \(k\) from \(1\) to \(\infty\), we easily obtain the required inequality. The lemma is proved.

Proof of Theorem 1. It is clear that it is enough to prove that at any point \(x\in V\) there is a neighborhood \(U\) such that \(u\in W_p^2(U)\) and \(Lu=-f\).

It follows from Lemma 3 that there exist a constant \(C\) and two neighborhoods \(U_1,U_2\) of the point \(x\) such that \(U_1\subset U_2\subset V\) and, for any function \(v\in W_p^2(U_2)\),

\[ \|v\|_{W_p^2(U_1)} \leq C\bigl(\|Lv\|_{L_p(U_2)}+\|v\|_{B(\partial U_2)}\bigr). \tag{7} \]

Moreover, one may assume that \(\partial U_2\) is twice continuously differentiable. We shall show that \(u\in W_p^2(U_1)\).

Denote by \(\tau\) the first exit time of \(X\) from \(U_2\), and apply Dynkin’s lemma to \(u\) and \(U=U_2\). By Lemma 2, \(R(U_2)f\in W_p^2(U_1)\) and \(LR(U_2)f=-f\). Hence, to prove the theorem it is enough to show that \(M_yu(x_\tau)\in W_p^2(U_1)\) and \(LM_yu(x_\tau)=0\).

First suppose that the function \(u(x)\) is continuous in the domain \(V\). Choose a sequence \(\{u_n;\ n\geq 1\}\subset C^2(V)\) such that \(\|u_n-u\|_{B(\partial U_2)}\to 0\) as \(n\to\infty\). As already mentioned, for each \(n\) the representation

\[ u_n(y)=-R(U_2)Lu_n(y)+M_yu_n(x_\tau) \tag{8} \]

is valid.

It follows that \(v_n(y)\equiv M_yu_n(x_\tau)\in W_p^2(U_2)\). Moreover, by Lemma 2 and (8), \(Lv_n=0\). Applying estimate (7), we obtain that there exists some function \(v\in W_p^2(U_1)\) to which the sequence \(\{v_n;\ n\geq 1\}\) converges in \(W_p^2(U_1)\), and hence uniformly. Finally, since
\[ |M_yu_n(x_\tau)-M_yu(x_\tau)|\leq \|u_n-u\|_{B(\partial U_2)}, \]
we have \(v(y)=M_yu(x_\tau)\), and
\[ LM_yu(x_\tau)=\lim_{n\to\infty}Lv_n(y)=0. \]

It remains to verify that \(u\in C(V)\). \(R(U_2)f\in W_p^2(U_2)\), and therefore the function \(R(U_2)f(x)\) is continuous at the point \(x\). Further, it is proved in [6] that \(X\) is a strongly Feller process; consequently, by Theorem 13.1 of [1], \(M_xu(x_\tau)\) is also continuous at the point \(x\). In view of the arbitrariness of the point \(x\), this means precisely that \(u\) is continuous in \(V\). The theorem is proved.

Corollary. There exists a constant \(C\), independent of \(f\) and \(\varphi\), such that

\[ \|u\|_{W_p^2(V')} \leq C\bigl(\|f\|_{L_p(V)}+\|\varphi\|_{B(\partial V)}\bigr). \]

Proof. Consider the mapping \(R\) which takes the point \((f,\varphi)\) of the space \(L_p(V)\times B(\partial V)\) to the point \(u\) of the space \(W_p^2(V')\) by formula (1). Clearly this mapping is closed. Therefore the assertion of the corollary follows from Theorem 11.2.4 of [9], according to which \(R\) is continuous.

The following theorem is, in a certain sense, the converse of Theorem 1.

Theorem 2. Let the function \(u(x)\) be continuous in \(\overline V\) and let \(u \in W_p^2(V')\)

\[ \left(p>\frac{l}{2}\right) \]

for every subdomain \(V'\) of the domain \(V\) such that \(\overline{V'} \subset V\). Suppose also that \(-f=Lu \in L_p(V)\). Then formula (1) holds for \(u(x)\).

Proof. We first prove that (1) is valid for smooth domains in the case when \(u \in W_p^2(V)\). To this end, choose a sequence of smooth functions \(\{u_n;\ n \geqslant 1\}\) such that \(\|u_n-u\|_{W_p^2(V)} \to 0\) as \(n \to \infty\). By the Sobolev embedding theorem [5], \(\|u_n-u\|_{B(V)} \to 0\) as \(n \to \infty\). Finally, since formula (1) is valid for \(u_n\), passing to the limit as \(n \to \infty\) and using Lemma 1, we obtain the required equality for \(u(x)\).

In the general case, take a sequence of smooth domains \(\{V_n;\ n \geqslant 1\}\) such that \(V_n \Subset V_{n+1}\) and

\[ \bigcup_{n=1}^{\infty} V_n = V . \]

According to what was said above, for each \(n\) and \(x \in V_n\),

\[ u(x)=M_x \int_0^\infty \chi_{[0,\tau_n)}(t) f(x_t)\,dt + M_x u(x_{\tau_n}), \tag{9} \]

where \(\tau_n\) is the first exit time from \(V_n\), and \(\chi_{[0,s)}(t)\) is the characteristic function of the half-interval \([0,s)\). Let us pass in this formula to the limit as \(n \to \infty\). It is not hard to prove that \(\tau_n \to \tau\) as \(n \to \infty\) (a.s. \(P_x\)) and \(\tau_n \leqslant \tau\) for \(n \geqslant 1\). In view of the continuity of the process \(X\), \(x_{\tau_n} \to x_\tau\) as \(n \to \infty\) on the set

\[ A=\{\omega:\lim_{n\to\infty}\tau_n=\tau<\zeta\}\cup \bigcup_n \{\omega:\tau_n=\zeta\}. \]

Moreover,

\[ \chi_{[0,\tau_n)}(t)f(x_t) \to \chi_{[0,\tau)}(t)f(x_t) \]

as \(n \to \infty\) almost everywhere with respect to the measure \(P_x \times \lambda\), where \(\lambda\) is Lebesgue measure on \([0,\infty)\),

\[ \left|\chi_{[0,\tau_n)}(t)f(x_t)\right| \leqslant \chi_{[0,\tau)} |f(x_t)| \]

and

\[ M_x \int_0^\infty \chi_{[0,\tau)}(t)|f(x_t)|\,dt = R(V)|f|(x)<\infty . \]

Thus, if we prove that \(P_x(A)=1\), then the proof of the theorem can be completed by passing to the limit in (9).

Substitute \(u(x)\equiv 1\) in (9) and pass to the limit as \(n \to \infty\). Then, observing that

\[ M_x 1(x_{\tau_n})=P_x\{\tau_n<\zeta\}\to P_x\{\tau<\zeta\}+1-P_x(A) = M_x 1(x_\tau)+1-P_x(A) \]

as \(n \to \infty\), we obtain

\[ 1=-R(V)L1(x)+M_x1(x_\tau)+1-P_x(A). \]

However, for \(1\) formula (1) is valid, and therefore \(P_x(A)=1\). The theorem is proved.

Corollary. For the function \(u\), the estimate of the corollary to Theorem 1 is valid, where \(f=-Lu\), \(\varphi(x)=u(x)\) for \(x \in \partial V\).

Theorem 2 would be the converse of Theorem 1 if Theorem 1 asserted the continuity of \(u(x)\) in \(\overline V\). It is clear, however, that, generally speaking, this is false. In § 2 we shall consider those cases when \(u(x)\) is nevertheless continuous at boundary points.

§ 2. CONTINUITY OF THE GENERALIZED SOLUTION AT BOUNDARY POINTS

We shall call a point \(a \in \partial V\) regular (for the operator \(L\) and the domain \(V\)) if, for every \(\varphi \in B(\partial V)\) continuous at the point \(a\),

\[ \lim_{x\to a} M_x \varphi(x_\tau)=\varphi(a), \]

where it is assumed that \(x\), as it tends to \(a\), remains in the domain \(V\)*) . From Theorem 2 it follows easily that our notion of regularity of a point coincides with the notion introduced in [10] in the case of sufficiently smooth coefficients of \(L\).

Let us derive two criteria for regularity of a point. In [3] the process \(X\) corresponding to the operator \(L(x)\) was constructed from the process \(X_1\) corresponding to \(L(x)+c(x)\) by means of killing with killing density \(c(x)\). Hence, and from item 2 of [7], it follows easily that the regular points for the operators \(L(x)\) and \(L(x)+c(x)\) coincide, and for regularity of a point \(a\) it is necessary and sufficient that \(P_a\{\tau' >0\}=0\), while for irregularity it is necessary and sufficient that \(P_a\{\tau'>0\}=1\)**, where \(\tau'\) is the time of the first exit from \(V\) after zero, i.e. \(\tau'=\inf\{t:t>0,\ x_t\notin V\}\), if there exist \(t>0\) for which \(x_t\notin V\), and \(\tau'=\zeta\) otherwise.

This reasoning shows that, without loss of generality, one may assume \(c(x)\equiv 0\).

Further, by Theorems 13.6 and 13.7 of [1] we obtain that, for regularity of the point \(a\), it is necessary and sufficient that

\[ \lim_{x\to a} M_x \tau=\lim_{x\to a}\int_V g(x,y)\,dy=0. \]

The following lemma will allow us, in investigating the question of boundary values, to assume \(f(x)\equiv 0\).

Lemma 4. Let \(a\) be a regular point, \(f\in L_p(V)\) \(\left(p>\frac{l}{2}\right)\), and let \(\varphi\) be a bounded function continuous at the point \(a\). Suppose that there exists a neighborhood \(U\) of the point \(a\) in which \(f(x)\) is bounded. Then the function \(u(x)\) defined by formula (1) satisfies the relation \(\lim\limits_{x\to a}u(x)=\varphi(a)\).

Proof. By Dynkin’s lemma,
\[ u(x)=R(U\cap V)f(x)+M_x\overline{\varphi}(x_{\tau_1}). \]
From the first criterion of regularity it follows that \(a\) will be regular also for the domain \(V\cap U\). Therefore
\[ \lim_{x\to a}M_x\overline{\varphi}(x_{\tau_1})=\varphi(a). \]
Using the boundedness of \(f(y)\), by the second criterion of regularity we obtain

\[ \overline{\lim}_{x\to a} R(U\cap V)f(x)\leq \|f\|_{B(U)}\,\overline{\lim}_{x\to a} R(U\cap V)1(x)\leq \]

\[ \leq \|f\|_{B(U)}\lim_{x\to a} R(V)1(x)=0. \]

The lemma is proved.

Our further plan is as follows. First it is proved that, under certain conditions, the regular points for the operator \(L\) are the same as for the Laplace operator (Theorem 4). In doing so, results of [7] and one estimate of the Green function of the operator \(L\) (Theorem 3) are used.

*) In what follows, when this causes no confusion, we shall not make this reservation.

**) By the zero-one law, \(P_a\{\tau'>0\}\) is equal to 0 or 1.

Then, in the general case, we shall prove that if the points of the boundary can be touched from outside the domain by an open cone, then it is regular (Theorem 5).

Introduce some notation and functions. If \(Q\) is a matrix of order \(l\times l\), then \(\operatorname{tr} Q\) denotes the trace of this matrix. Let \(A(x)=(a_{ij}(x))\); \(C(x)\) be the positive symmetric square root of the matrix \(A^{-1}(x)\);

\[ \overline w(x,h)=\max_{\|\lambda\|=1}\max_{\|y-x\|\le h} \bigl(C(x)C^{-2}(y)C(x)\lambda,\lambda\bigr) =\max_{\lambda,\|y-x\|\le h} \frac{(A(y)\lambda,\lambda)}{(A(x)\lambda,\lambda)}, \]

\[ \underline w(x,h) \]

be the minimum of the same expression over the same set;

\[ \overline b(x,h)=\mu^{-2}h^{-1}\left[ l\bigl(\overline w(x,h\mu^{-1/2})-\underline w(x,h\mu^{-1/2})\bigr) +Mh\mu^{-1/2}\right], \]

\[ \underline b(x,h)=-\overline b(x,h). \]

Suppose that for some neighborhood \(U\) of the point \(a\) and all \(x\in U\)

\[ \int_0^1 h^{-1}\bigl(\overline w(x,h)-\underline w(x,h)\bigr)\,dh\le \mathrm{const}<\infty . \tag{10} \]

We then introduce two more functions

\[ \overline g_R(x,y)= \int_{\|C(x)y\|}^{R} t^{1-l}\exp\left(\int_0^t \overline b(x,h)\,dh\right)\,dt, \]

\[ \underline g_R(x,y) \]

is obtained if, in this expression, \(\overline b\) is replaced by \(\underline b\).

In view of the importance of condition (10), let us note that it is satisfied if

\[ \int_0^1 h^{-1}\omega(h)\,dh<\infty . \]

Indeed, this follows from the obvious inequalities

\[ 0\le \overline w(x,h)-1 = \max_{\lambda,\|y-x\|\le h} \left[ \frac{(A(y)\lambda,\lambda)}{(A(x)\lambda,\lambda)}-1 \right] \le C\omega(h), \]

\[ 0\le 1-\underline w(x,h) = \max_{\lambda,\|y-x\|\le h} \left[ 1-\frac{(A(y)\lambda,\lambda)}{(A(x)\lambda,\lambda)} \right] \le C\omega(h), \]

where \(C\) depends only on \(\mu\) and \(l\).

The following lemma can be proved by means of simple computations.

Lemma 5. If \(\lambda=C(x)(y-x)\), then

\[ \|\lambda\|^{-2}\bigl(C(x)A(y)C(x)\lambda,\lambda\bigr)\, \overline b(y,\|\lambda\|) \ge \|\lambda\|^{-1}l\|\lambda\|^{-2} \bigl(C(x)A(y)C(x)\lambda,\lambda\bigr) -\operatorname{tr}\bigl(C(x)A(y)C(x)\bigr) -\bigl(C(x)\lambda,b(y)\bigr). \]

If \(y\in U\) and \(\|C(y)(x-y)\|<R\), then

\[ L(x)\,\overline g_R(y,x-y)\le 0\le L(x)\,\underline g_R(y,x-y), \]

where the operator \(L(x)\) is applied with respect to the variable \(x\).

Theorem 3. Let \(R\) be such that \(U\) is contained in some ball of radius \(2^{-1}\sqrt{\mu}\,R\), and let \(G_R\) be the union of all such balls, \(p>l/2\), \(l\ge 3\),

\[ V\subset \overline V\subset G_R. \]

Then, if \(f\in L_p(U)\), \(f\ge 0\), and \(f=0\) outside \(U\), then for \(x\in V\)

\[ \overline u(x)\equiv \int \overline g_R(y,x-y)f(y)\,dy \ge C^{-1}R(V)f(x)+M_x\overline u(x_\tau), \]

\[ \underline u(x)\equiv \int \underline g_R(y,x-y)f(y)\,dy \le CR(V)f(x)+M_x\underline u(x_\tau), \]

where \(C\) depends only on \(\mu\) and \(l\).

Proof. It is clear that \(f\) may be assumed to be a bounded smooth function. Next, in this proof we redefine \(\bar b(x,h)\), leaving it unchanged if \(x \in U\), and setting it equal to zero if \(x \notin U\). Obviously, the functions \(\bar u(x)\) and \(\underline u(x)\) are not changed thereby.

Let \(h(r)\) be a twice continuously differentiable function on \([0,\infty)\), equal to zero on \([0,1]\) and to one outside \([0,2]\). Put \(h_\varepsilon(x)=h\left(\frac{\|x\|}{\varepsilon}\right)\) and

\[ \bar u_\varepsilon(x)=\int \bar g_R(y,x-y)h_\varepsilon(x-y)f(y)\,dy . \]

It is clear that \(\bar u_\varepsilon(x)\) has two derivatives in \(G_R\). Therefore
\(\bar u_\varepsilon(x)=-R(V)L\bar u_\varepsilon(x)+M_x\bar u_\varepsilon(x_\tau)\).
Passing in this equality to the limit as \(\varepsilon\to 0\), we obtain
\(\bar u(x)=-\lim_{\varepsilon\to 0}R(V)L\bar u_\varepsilon(x)+M_x\bar u(x_\tau)\).
Thus, in order to prove the theorem for \(\bar u(x)\), it is sufficient to establish that

\[ I=\lim_{\varepsilon\to 0}R(V)L\bar u_\varepsilon(x)\leq -C^{-1}R(V)f(x). \tag{11} \]

We have

\[ L(x)\bar u_\varepsilon(x)= \int f(y)L(x)\left[\bar g_R(y,x-y)h_\varepsilon(x-y)\right]\,dy . \]

Applying Lemma 5 and making the change of variables \(y=x-\varepsilon z\) in the last integral, it is easy to obtain that
\(I\leq \lim_{\varepsilon\to 0} R(V) I(\varepsilon,x)\), where

\[ I(\varepsilon,x)=\int f(x-\varepsilon z) \left[ 2\left(A(x)Dh_1(z),\varepsilon^{l-1}D\bar g_R(x-\varepsilon z,\varepsilon z)\right) +\varepsilon^{l-2}\bar g_R(x-\varepsilon z,\varepsilon z)L_1(x)h_1(z) \right]\,dz . \]

Here \(L_1(x)\) is the principal part of the operator \(L(x)\); \(Dh_1(z)\), \(D\bar g_R(y,z)\) are the gradients of \(h_1(z)\) and \(\bar g_R(y,z)\) as functions of \(z\).

Let \(F_\varepsilon(x-\varepsilon z,z)\) be obtained from the integrand of the last integral by replacing \(A(x)\) and \(L_1(x)\), respectively, by \(A(x-\varepsilon z)\) and \(L_1(x-\varepsilon z)\). It is not hard to see that \(F_\varepsilon(x,z)\) is bounded for all \(x,z,\varepsilon\), and

\[ \lim_{\varepsilon\to 0}R(V)I(\varepsilon,x) = \lim_{\varepsilon\to 0}R(V)\int F_\varepsilon(x-\varepsilon z,z)\,dz . \tag{12} \]

Next, setting \(\bar g_V(x,y)\equiv g(x,y)=0\) for \(y \notin V\), we obtain

\[ \left| R(V)\int F_\varepsilon(x-\varepsilon z,z)\,dz - R(V)\int F_\varepsilon(x,z)\,dz \right| \leq C\int_{\|z\|\leq 2} dz \int \left|g(x,y+\varepsilon z)-g(x,y)\right|\,dy . \]

Therefore equality (12) remains valid if in it \(F_\varepsilon(x-\varepsilon z,z)\) is replaced by \(F_\varepsilon(x,z)\). \(F_\varepsilon(x,z)\) is the sum of two terms. Transform the integral of the second with respect to \(dz\) by integration by parts; then

\[ \lim_{\varepsilon\to 0}R(V)I(\varepsilon,x) = \lim_{\varepsilon\to 0}R(V)f(x)\int \left(A(x)Dh_1(z),\varepsilon^{l-1}D\bar g_R(x,\varepsilon z)\right)\,dz = \]

\[ = -R(V)f(x)\int \left(A(x)Dh_1(z),D\|C(x)z\|\right)\|C(x)z\|^{1-l}\,dz . \]

From these equalities, (11) is obtained by a simple estimate of the last integral.

The theorem for \(\overline u(x)\) has been proved. For \(\underline u(x)\) the proof is analogous.

Corollary. Let \(X_\Delta\) be the process corresponding to the Laplace operator (the Wiener process), and let \(R_\Delta(U)\) be the operator defined from \(X_\Delta\) in the same way as \(R(U)\) from \(X\). Then at the point \(a\) there exists a spherical neighborhood \(S(r)\) of radius \(r\) such that, for any positive function \(f(x)\) equal to zero outside \(S(r)\), for \(x \in S(r)\),
\[ C_1^{-1} R_\Delta(V) f(x) \leq R(V) f(x) \leq C_1 R_\Delta(V) f(x), \]
where \(C_1\) does not depend on \(x\) or on \(f\).

Proof. Let us first note that, for the Wiener process, the function whose existence is asserted in Lemma 1 is the Green function of the Laplace operator in the domain \(V\). From the known estimates for the Green function \(g_\Delta(x,y)\) of the Laplace operator in the domain \(V\) it follows that there is a constant \(C_2\) such that, for all sufficiently small \(r\),
\[ \underline g_R(y,x-y) \geq C_2 g_\Delta(x,y), \]
if \(x,y \in S(r)\).

Further, it is not difficult to see that \(r\) can be chosen so that
\[ \underline g_R(y,x-y) \geq 2 \underline g_R(y,z-y) \]
for \(x,y \in S(r)\), \(z \in \partial V\). Then
\[ R(V) f(x) \geq (2C)^{-1} \int \underline g_R(y,x-y) f(y)\,dy \geq C_3 R_\Delta(V) f(x). \]
The left inequality has been proved. The proof of the right inequality is analogous.

We have now prepared everything necessary to prove the main result of this section.

Theorem 4. Let condition (10) be satisfied for some neighborhood \(U\) of the point \(a\). Then \(a\) is regular for \(L\) if and only if it is regular for the Laplace operator*).

Proof. In view of the corollary to Theorem 3 and item 3 of [7]**) it is enough, in order to prove this theorem for \(l \geq 3\), to show that, for all \(\varepsilon>0\),
\[ \lim_{t\to 0} t^{-1}\sup_{x\in U} P_x\{\, \|x_t-x\|>\varepsilon \,\}=0. \tag{13} \]
By Chebyshev’s inequality,
\[ P_x\{\,\|x_t-x\|>\varepsilon\,\} \leq \varepsilon^{-4} M_x\|x_t-x\|^4. \]
Further, in [3] it is proved that
\[ M_x\|x_t-x\|^4 \leq C(t^2+t^4), \]
where \(C\) does not depend on \(x\). Consequently, condition (13) is fulfilled.

Now let \(l<3\). We regard \(E_l\) as the plane
\[ \{\overline x=(x,x_{l+1},x_{l+2})=(x_1,\ldots,x_{l+2}): x_{l+1}=0,\ x_{l+2}=0\} \]
in \(E_{l+2}\). Let
\[ L_1(\overline x) = L(x) + \frac{\partial^2}{\partial x_{l+1}^2} + \frac{\partial^2}{\partial x_{l+2}^2}, \]
and let \(\overline X\) be the process in \(E_{l+2}\) corresponding to \(L_1(\overline x)\). It is not difficult to prove that the process obtained from \(\overline X\) by orthogonal projection of the trajectories of \(\overline X\) onto \(E_l\) is equivalent to the process \(X\). Hence, by the first criterion of regularity of a point, we obtain that the point \(a\) is regular for the domain \(V\subset E_l\) and the operator \(L(x)\) if and only if \((a,0,0)\) is regular for the domain \(V\times(-1,1)^2\) and the operator \(L_1(\overline x)\). Finally, since condition (10) is satisfied for \(L_1(\overline x)\) and \(l+2\geq 3\), \((a,0,0)\) is regular for \(L_1(\overline x)\) if and only if it is regular for the Laplace operator (in \(E_{l+2}\)). The theorem is proved.

*) This theorem, under stronger assumptions on \(L\), has been proved in many papers; see, for example, [10].

**) Roughly speaking, in item 3 of [7], just as in [8], the coincidence of the notion of regularity for different operators is derived from estimates for the Green function.

In the case where condition (10) is not satisfied, generally speaking it apparently cannot be asserted that the regular points for different operators will coincide. However, the following holds.

Theorem 5. Suppose the point \(a\) can be touched from outside the domain by an open cone \(K\); then it is regular.

Proof. Let \(a=0\), and let \(\tau'_1\) be the first time after zero at which \(K\) is hit. Clearly,
\(P_0\{\tau' > 0\} \leq P_0\{\tau'_1 > 0\}\), and hence, by the first regularity criterion, it is enough to prove that \(P_0\{\tau'_1 > 0\}<1\).

As in Lemma 13.3 of [2],

\[ P_0\{\tau'_1>t\}\leq 1-P_0\{x_t\in K\}. \tag{14} \]

Let \(\xi_s\) be an \(l\)-dimensional Wiener random function. From the estimate for \(R(V)f\) from Lemma 1 and from [3] it is not difficult to obtain that there exists a solution of the stochastic integral equation

\[ y_t=\int_0^t C^{-1}(y_t)\,d\xi_t+\int_0^t b(y_t)\,dt^{*}), \]

such that, for all \(t\),

\[ P_0\{x_t\in K\}=P\{y_t\in K\}=P\{t^{-1/2}y_t\in K\}. \]

Further,
\[ t^{-1/2}y_t=C^{-1}(0)t^{-1/2}\xi_t+\alpha_t, \]
where
\[ \alpha_t=t^{-1/2}\int_0^t [C^{-1}(y_t)-C^{-1}(0)]\,d\xi_t +t^{-1/2}\int_0^t b(y_t)\,dt . \]

From the properties of the stochastic integral we obtain

\[ M\alpha_t^2\leq 2t^{-1}\int_0^t M\|C^{-1}(y_t)-C^{-1}(0)\|^2\,dt+2M^2t^{**}) \]

and hence \(M\alpha_t^2\to 0\) as \(t\to 0\). Now, since the distribution of \(t^{-1/2}\xi_t\) does not depend on \(t\), we have
\[ \lim_{t\to 0} P\{t^{-1/2}y_t\in K\} = P\{C^{-1}(0)\xi_1\in K\}>0. \]
Therefore, from (14),
\[ P_0\{\tau'_1>0\}=\lim_{t\to 0}P_0\{\tau'_1>t\}<1, \]
as was required.

In conclusion I would like to express my gratitude to A. Martin-Löf for a conversation which served as my starting point in finding the functions \(\bar g_R\) and \(\underline g_R\), and also to V. A. Kondrat'ev, who reviewed the work in manuscript.

References

1. Dynkin E. B. Markov Processes. Moscow, Fizmatgiz, 1963.
2. Tanaka H. Mem. Fac. Sci. Kyushu Univ., Ser. A, 18, 1, 89—103, 1964.
3. Krylov N. V. Probability theory and its applications, 11, No. 3, 424—443, 1966.
4. Agmon S., Douglis A., Nirenberg L. Estimates near the boundary of solutions of elliptic partial differential equations under general boundary conditions. Moscow, IL, 1962.

*) See Ch. 7 [1].

**) Here the norm of the matrix \((a_{ij})\) is defined as
\[ \sqrt{\sum_{i,j=1}^{l} a_{ij}^{2}}. \]

1. Sobolev S. L. Some applications of functional analysis in mathematical physics. L., Publishing House of Leningrad University, 1950.

2. Krylov N. V. UMN, vol. XXI, issue 1, 177—179, 1966.

3. Krylov N. V. Probability Theory and Its Applications, 11, No. 4, 634—638, 1966.

4. Weinberger H. F., Littman W., Stampacchia G. Collection of translations “Mathematics,” 9:2, 1965, pp. 72—97.

5. Dunford N., Schwartz J. T. Linear Operators. M., IL, 1962.

6. Oleinik O. A. Matem. sb., new series, 24, No. 1, 1949, pp. 3—14.

Received by the editors
29 November 1965

Moscow State University
named after M. V. Lomonosov