UDC 517.948

MATHEMATICS

M. N. FELLER

ON THE EQUATION \(\Delta U[x(t)] + P[x(t)]U[x(t)] = 0\) IN FUNCTION SPACE

(Presented by Academician A. Yu. Ishlinskii on 16 IV 1966)

The Laplacian of a functional \(\Delta U[x(t)]\)—the continual analogue of the Laplace operator of a function of a finite number of variables—was defined by P. Lévy \((^{1})\). Laplace and Poisson equations with such Laplacians were studied by P. Lévy, E. M. Polishchuk \((^{2})\), and the author \((^{3,4})\). In this note we consider the boundary-value problem of the “generalized Laplace equation”

\[ \Delta U[x(t)] + P[x(t)]U[x(t)] = 0 \tag{1} \]

for a domain \(\Omega \cup \Gamma\) in the space \(C\), defined by the inequality \(S[x(t)] \leqslant 1\) \((x(t) \in \Omega\), if \(S[x(t)] < 1\); \(x(t) \in \Gamma\), if \(S[x(t)] = 1)\); \(S[x(t)]\) is a twice functionally differentiable functional such that its first and second variations have normal form, and \(\Delta S[x(t)]\) is a constant positive nonzero number; \(C\) is the space of functions \(x(t)\) continuous on \([0,1]\) \((x(0)=0)\), with Wiener measure \((^{5})\).

1. Let \(P[x(t)]\) be a functional of finite degree \((^{6})\); \(P_N[x(t)]\) the partial sum of its expansion in the Fourier–Hermite functionals \((^{7})\); \(B_{m_1\ldots m_N}\) its Fourier–Hermite coefficients;

\[ \Psi_{m_1\ldots m_N}[x(t)] = \prod_{i=1}^{N} H_{m_i}\left[\int_{0}^{1}\chi_i(t)\,dx(t)\right] \]

are the Fourier–Hermite functionals \((m_i=0,1,2,\ldots;\ N=1,2,\ldots)\); \(H_m(u)\) are partially normalized Hermite polynomials \((m=0,1,2,\ldots)\); \(\chi_1(t)\equiv \chi_0^{(0)}(t)\), \(\chi_i(t)\equiv \chi_n^{(k)}(t)\) for \(i=2^n+k\) \((n=0,1,2,\ldots;\ k=1,2,\ldots,2^n)\), \(\chi_0^{(0)}(t)\), \(\{\chi_n^{(k)}(t)\}\) are the Haar system of functions. Let

\[ \Psi^{*}_{m_1\ldots m_N}[x(t),y_1(\tau),\ldots,y_N(\tau)] = \]

\[ = \prod_{i=1}^{N} H_{m_i} \left[ \int_{0}^{1}\chi_i(t)\,dx(t) +2(1-S[x])^{1/2}(\Delta S[x])^{-1/2}\,\widetilde y_i(\tau) \right], \]

\[ \widetilde y_1(\tau)\equiv y_1(\tau),\quad \widetilde y_i(\tau)\equiv \widetilde y_n^{(k)}(\tau),\quad \widetilde y_n^{(k)}(\tau) = \sqrt{2^n}\left(2z_{(2k-1)/2^{n+1}}(\tau)-z_{(2k-2)/2^{n+1}}(\tau)-z_{2k/2^{n+1}}(\tau)\right), \]

\[ z_0(\tau)=0,\qquad z_{p_q}(\tau)=y_q(\tau),\qquad (p_q=1,\;1/2,\;1/4,\;3/4,\ldots;\ q=1,2,\ldots,N). \]

Lemma 1. The functional

\[ \Phi_{m_1\ldots m_N,N}[x(t)] = \int_{C_N} \Psi^{*}_{m_1\ldots m_N} [x(t),y_1(1),\ldots,y_N(1)] \times \]

\[ \times \exp\left\{ \frac{1-S[x]}{\Delta S[x]} \int_{0}^{1} \sum_{\mu_1,\ldots,\mu_N=0}^{N} B_{\mu_1\ldots\mu_N} \Psi^{*}_{\mu_1\ldots\mu_N} [x(t),y_1(s),\ldots,y_N(s)]\,ds \right\} \,dw_{y_1}\cdots dw_{y_N}, \]

where the integral is understood as an \(N\)-fold Wiener integral on the product space \(C\), satisfies in \(\Omega\) the equation

\[ \Delta U[x]+P_N[x]U[x]=0. \]

Indeed, having found the second variation and then computed the Laplacian of the functional \(\Phi_{m_1\ldots m_N,N}[x]\), we obtain

\[ \begin{aligned} \Delta \Phi_{(m),N}[x] = \int_{C^N} \Bigg\{& \sum_{i=1}^{N}\left[2^{n_i+1}\sqrt{m_i(m_i-1)}\,a_i\Psi^*_{(m)-2_i}(1)\right.\\ &\left.-(1-S)^{-1/2}(\Delta S)^{1/2}\sqrt{2m_i}\,\widetilde y_i(1)\Psi^*_{(m)-1_i}(1)\right.\\ &\left.+\sum_{\substack{l=1\\ l\ne i}}^{N}\sqrt{2^{n_i+n_l+2}m_i m_l}\,a_{il}\Psi^*_{(m)-1_i-1_l}(1)\right]\\ &+2(1-S)(\Delta S)^{-1}\sqrt{2^{n_i+1}m_i}\,b_i\Psi^*_{(m)-1_i}(1)\\ &\qquad\times\left[\int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)}\sum_{l=1}^{N}\sqrt{2^{n_l+1}\mu_l}\,b_l\Psi^*_{(\mu)-1_l}(s)\,ds\right] -\Psi^*_{(m)}(1)\int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)}\Psi^*_{(\mu)}(s)\,ds\\ &+(1-S)(\Delta S)^{-1}\Psi^*_{(m)}(1) \int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)}\sum_{i=1}^{N} \left[2^{n_i+1}\sqrt{\mu_i(\mu_i-1)}\,a_i\Psi^*_{(\mu)-2_i}(s)\right.\\ &\left.\qquad\qquad\qquad\qquad -(1-S)^{-1/2}(\Delta S)^{1/2}\sqrt{2\mu_i}\,\widetilde y_i(s)\Psi^*_{(\mu)-1_i}(s)\right.\\ &\left.\qquad\qquad\qquad\qquad +\sum_{\substack{l=1\\ l\ne i}}^{N}\sqrt{2^{n_i+n_l+2}\mu_i\mu_l}\,a_{il}\Psi^*_{(\mu)-1_i-1_l}(s)\right]ds\\ &+(1-S)^2(\Delta S)^{-2}\Psi^*_{(m)}(1) \left[\int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)} \sum_{i=1}^{N}\sqrt{2^{n_i+1}\mu_i}\,B_i\Psi^*_{(\mu)-1_i}(s)\,ds\right]^2 \Bigg\}E_N\,dw y_1\ldots dw y_N, \end{aligned} \tag{2} \]

where
\[ \Psi^*_{m_1\ldots m_N}(\tau)=\Psi^*_{m_1\ldots m_N}[x(t),y_1(\tau),\ldots,y_N(\tau)],\qquad (m)=(m_1,\ldots,m_N), \]

\[ (m)-q_i=(m_1,\ldots,m_{i-1},m_i-q,m_{i+1},\ldots,m_N), \]
and \(a_i,a_{il},b_i\) are the sums of the coefficients at the corresponding \([\delta x]^2\),

\[ E_N=\exp\left\{\frac{1-S}{\Delta S}\int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)}\Psi^*_{(\mu)}(s)\,ds\right\}. \]

Computing, by P. Cameron’s theorem \((8)\), the Wiener integral of the first partial variation with respect to \(y_i\) of the functional
\[ (i-S)^{-1/2}(\Delta S)^{1/2}\sqrt{2m_i}\Psi^*_{(m)-1_i}(1)E_N, \]
we have

\[ \begin{aligned} &(i-S)^{-1/2}(\Delta S)^{1/2} \int_{C^N}\sum_{i=1}^{N}\sqrt{2m_i}\,\widetilde y_i(1)\Psi^*_{(m)-1_i}(1)E_N\,dw y_1\ldots dw y_N\\ &=\int_{C^N}\sum_{i=1}^{N}\Bigg\{ 2^{n_i+1}\sqrt{m_i(m_i-1)}\,a_i\Psi^*_{(m)-2_i}(1) +\sum_{\substack{l=1\\ l\ne i}}^{N}\sqrt{2^{n_i+n_l+2}m_i m_l}\,a_{il}\Psi^*_{(m)-1_i-1_l}(1)\\ &\qquad +(1-S)(\Delta S)^{-1}\sqrt{2^{n_i+1}m_i}\,b_i\Psi^*_{(m)-1_i}(1) \int_0^1 s\sum_{(\mu)=0}^{N}B_{(\mu)} \sum_{l=1}^{N}\sqrt{2^{n_l+1}\mu_l}\,b_l\Psi^*_{(\mu)-1_l}(s)\,ds \Bigg\}\\ &\qquad\times E_N\,dw y_1\ldots dw y_N, \end{aligned} \tag{3} \]

and by M. Ouchard’s theorem \((9,10)\)—for the partial variational derivative of the functional
\[ (1-S)^{1/2}(\Delta S)^{-1/2}\Psi^*_{(m)}(1)\sqrt{2\mu_i}\Psi^*_{(\mu)-1_i}(s)E_N \]
with respect to \(y_i\)—we obtain

\[ \begin{aligned} &(1-S)^{1/2}(\Delta S)^{-1/2} \int_{C^N}\Psi^*_{(m)}(1) \left[\int_0^1\sum_{(\mu)=0}^{N}B_{(\mu)} \sum_{i=1}^{N}\sqrt{2\mu_i}\,\widetilde y_i(s)\Psi^*_{(\mu)-1_i}(s)\,ds\right]E_N\,dw y_1\ldots dw y_N\\ &=\int_{C^N}\Bigg\{ -2(1-S)^2(\Delta S)^{-2}\Psi^*_{(m)}(1) \int_0^s\sum_{(\mu)=0}^{N}B_{(\mu)} \sum_{i=1}^{N}\sqrt{2^{n_i+1}\mu_i}\,b_i\Psi^*_{(\mu)-1_i}(s)\\ &\qquad\times \int_1^s\sum_{(\mu)=0}^{N}B_{(\mu)} \sum_{l=1}^{N}\sqrt{2^{n_l+1}\mu_l}\,b_l\Psi^*_{(\mu)-1_l}(v)\,dv\,ds + \end{aligned} \]

\[ \begin{aligned} &+(1-S)(\Delta S)^{-1}\Psi^*_{(m)}(1)\left\{\int_0^1 s\sum_{(\mu)=0}^N B_{(\mu)}\sum_{i=1}^N \left[2^{n_i+1}\sqrt{\mu_i(\mu_i-1)}\,a_i\Psi^*_{(\mu)-2_i}(s)+\right.\right.\\ &\left.\left.+\sum_{\substack{l=1\\ l\ne i}}^N \sqrt{2^{n_i+n_l+2}\mu_i\mu_l}\,a_{il}\Psi^*_{(\mu)-1_i-1_l}(s)\right]\,ds\right\}\\ &+(1-S)(\Delta S)^{-1}\sum_{i=1}^N \sqrt{2^{n_i+1}}\,m_i b_i\Psi^*_{(m)-1_i}(1)\times\\ &\times\left\{\int_0^1 s\sum_{(\mu)=0}^N B_{(\mu)}\sum_{l=1}^N \sqrt{2^{n_l+1}}\mu_l b_l\Psi^*_{(\mu)-1_l}(s)\,ds\right\}E_N^2\,dwy_1\ldots dwy_N . \end{aligned} \tag{4} \]

Extending lemma \((10)\) to the \(N\)-dimensional case and using E. Ouchar’s theorem, it is not difficult to obtain the relation

\[ \begin{aligned} &\int_{C^N}\Psi^*_{(m)}(1)\int_0^1 \sum_{(\mu)=0}^N B_{(\mu)}\Psi^*_{(\mu)}(s)\,ds\,E_N\,dwy_1\ldots dwy_N = P_N[x]\Phi_{(m),N}[x]-\\ &-\int_{C^N}\left\{2(1-S)^2(\Delta S)^{-2}\Psi^*_{(m)}(1)\int_0^1(1-s)\sum_{(\mu)=0}^N B_{(\mu)}\sum_{i=1}^N \sqrt{2^{n_i+1}}\mu_i b_i\Psi^*_{(\mu)-1_i}(s)\times\right.\\ &\left.\times\int_1^s \sum_{(\mu)=0}^N B_{(\mu)}\sum_{l=1}^N \sqrt{2^{n_l+1}}\mu_l b_l\Psi^*_{(\mu)-1_l}(v)\,dv\,ds-\right.\\ &\left.-(1-S)(\Delta S)^{-1}\Psi^*_{(m)}(1)\int_0^1(1-s)\sum_{(\mu)=0}^N B_{(\mu)}\sum_{i=1}^N \left[2^{n_i+1}\sqrt{\mu_i(\mu_i-1)}\,a_i\Psi^*_{(\mu)-2_i}(s)+\right.\right.\\ &\left.\left.+\sum_{\substack{l=1\\ l\ne i}}^N \sqrt{2^{n_i+n_l+2}\mu_i\mu_l}\,a_{il}\Psi^*_{(\mu)-1_i-1_l}(s)\right]\,ds-\right.\\ &\left.-2(1-S)(\Delta S)^{-1}\sum_{i=1}^N \sqrt{2^{n_i+1}}\,m_i b_i\Psi^*_{(m)-1_i}(1)\times\right.\\ &\left.\times\int_0^1(1-s)\sum_{(\mu)=0}^N B_{(\mu)}\sum_{l=1}^N \sqrt{2^{n_l+1}}\mu_l b_l\Psi^*_{(\mu)-1_l}(s)\,ds\right\} E_N\,dwy_1\ldots dwy_N . \end{aligned} \tag{5} \]

Substituting now (3), (4), and (5) into (2), we obtain that \(\Phi_{(m),N}[x]= -P_N(x)\Phi_{(m),N}[x]\) in the domain \(\Omega\).

From Lemma 2 and Corollary 1 \((^3)\) it follows that

Lemma 2. If \(P[x(t)]<0\), \(\Omega_h\cup\Gamma_h\) is the closure of the set of functions

\[ x_h(t)=\frac{1}{2h}\int_{t-h}^{t+h} x(s)\,ds,\qquad x(t)\in\Omega\cup\Gamma,\qquad x(t)=0\quad \text{outside }[0,1], \]

then a functional \(V[x(t)]\), satisfying equation (1) in \(\Omega_h\) and continuous in \(\Omega_h\cup\Gamma_h\) for any \(h>0\), cannot attain a positive maximum and a negative minimum in \(\Omega_h\).

Corollary. If \(P[x(t)]\leq 0\), and the functionals \(V_1[x(t)]\) and \(V_2[x(t)]\) satisfy equation (1) in \(\Omega\) and \(\Omega_h\), are continuous in \(\Omega\cup\Gamma\) and \(\Omega_h\cup\Gamma_h\), and \(V_1|_{\Gamma}=V_2|_{\Gamma}=H[x]\), then \(V_1\equiv V_2\) in \(\Omega\).

2. Theorem 1. Suppose a finite-degree functional \(G[x(t)]\) is given; \(A_{m_1\ldots m_N}\) are its Fourier–Hermite coefficients; \(P[x(t)]\) is a finite-degree functional, \(P[x(t)]\leq 0\). Then in the domain \(\Omega\) there exists a unique solution of equation (1), coinciding with the given functional \(G[x(t)]\) on the surface \(\Gamma\), which is equal to

\[ U[x(t)]=\lim_{N\to\infty}\sum_{m_1,\ldots,m_N=0}^{N} A_{m_1\ldots m_N}\Phi_{m_1\ldots m_N,N}[x(t)] \tag{6} \]

for almost all \(x(t)\in\Omega\cup\Gamma\).

Let us now consider the solution of equation (1) in the space \(L_2(C)\).

Theorem 2. Let a functional \(G[x(t)] \in L_2(C)\) be given; let \(A_{m_1\ldots m_N}\) be its Fourier–Hermite coefficients; let \(P[x(t)]\) be a functional of finite degree, bounded below almost everywhere, \(P[x(t)] \leqslant 0\). Then in the domain \(\Omega\) there exists a unique solution \(U[x(t)] \in L_2(C)\)

\[ U[x(t)] = \underset{N\to\infty}{\mathrm{L.I.M.}}\, \sum_{m_1,\ldots,m_N=0}^{N} A_{m_1\ldots m_N}\, \Phi_{m_1\ldots m_N,N}[x(t)] \tag{7} \]

of equation (1), equal to the prescribed functional \(G[x(t)]\) on the surface \(\Gamma\).

The proof of the theorems follows from the properties of the functionals \(G[x(t)]\), \(P[x(t)]\), Lemma 1, and the corollary.

We note that, putting \(P[x(t)] \equiv 0\) in (6) and (7), we obtain the solution of the boundary-value problem for the “Laplace equation.” It coincides with the solution given in the author’s paper \((^3)\), if one computes the multiple Wiener integral of the functional concentrated at a point and then uses the Paley–Wiener formula \((^{11})\).

In conclusion I express my sincere gratitude to Yu. L. Daletskii for his great attention to the present work.

Ukrainian Scientific-Research Institute
of Mechanical Wood Processing

Received
8 IV 1966

CITED LITERATURE

\(^{1}\) P. Levy, Problèmes concrets d’analyse fonctionnelle, Paris, 1951.
\(^{2}\) E. M. Polishchuk, UMN, 19, no. 2 (116), 155 (1964).
\(^{3}\) M. N. Feller, Dop. AN URSR, 12, 1558 (1965).
\(^{4}\) M. N. Feller, Dop. AN URSR, 4, 426 (1966).
\(^{5}\) N. Wiener, Acta Math., 55, 117 (1930).
\(^{6}\) R. E. Graves, Proc. Am. Math. Soc., 4, 1, 95 (1953).
\(^{7}\) R. H. Cameron, W. T. Martin, Ann. Math., 48, 2, 385 (1947).
\(^{8}\) R. H. Cameron, Proc. Am. Math. Soc., 2, 6, 944 (1951).
\(^{9}\) M. Ovchar, Proc. Am. Math. Soc., 3, 3, 459 (1952).
\(^{10}\) R. H. Cameron, Ann. Math., 59, 3, 434 (1954).
\(^{11}\) H. Wiener, R. Paley, The Fourier Transform in the Complex Domain, Moscow, 1964.