L. I. KAMYNIN

ON LYAPUNOV THEOREMS FOR HEAT POTENTIALS

(Presented by Academician S. L. Sobolev on 30 VI 1964)

In potential theory, an important role is played by the profound investigations of A. M. Lyapunov \((^1)\) concerning the properties of various potentials (in particular, their behavior near the boundary) and the related solutions of a number of boundary-value problems for elliptic equations. In our note we present some theorems of Lyapunov type (Theorems 3–5) for heat potentials, establishing those conditions on the density and on the surface under which the heat potential of a simple layer, of a double layer, and the normal derivative of the heat potential of a simple layer possess prescribed smoothness.

As one of the applications of the Lyapunov theorems, we prove the existence of a \((2+\alpha)\)-smooth solution of the third boundary-value problem for a linear parabolic equation of the second order. The method of analytic continuation used here is essentially based on the \((2+\alpha)\) a priori estimate (up to the boundary) for solutions of the indicated problem, established by the author jointly with V. N. Maslennikova in \((^3)\).

§ 1. Definition. Let, for a closed surface \(\Gamma\) of \((n+1)\)-dimensional space \((x,t)=(x_1,x_2,\ldots,x_n;t)\), enclosed between the hyperplanes \(t=0\) and \(t=T>0\), there exist a constant \(R>0\) such that in every sphere \(S_R(x,t)\) with center at the point \((x,t)\in\Gamma\) and radius \(R\) the surface \(\Gamma\) admits a representation of the form

\[ x_i=g(x_1,x_2,\ldots,x_{i-1},x_{i+1},\ldots,x_n;t) \tag{1} \]

(without loss of generality we shall take \(i=n\)).

A. If \(g(x,t)\) in (1) is such that \(\partial^{2m-p}g/\partial t^p \partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\) \((p=0,1,\ldots,m)\) satisfies, with respect to \(t\), a Hölder condition with exponent \((1+\alpha)/2\), and \(\partial^{2m-p+1}g/\partial t^p \partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\) \((p=0,1,\ldots,m)\) satisfies, with respect to \(x\) and \(t\), Hölder conditions with exponents \(\alpha_1\) and \(\alpha/2\), respectively, then we shall say that the surface \(\Gamma\) is of type

\[ \mathcal{L}_{2m+1,\alpha_1,\alpha/2}^{m,1,(1+\alpha)/2} \quad (0<\alpha,\ \alpha_1\leq 1,\ m=0,1,2,\ldots). \]

B. If \(g(x,t)\) in (1) is such that \(\partial^{2m-p}g/\partial t^p \partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\) satisfies, with respect to \(x\) and \(t\), Hölder conditions with exponents \(\alpha_1\) and \(\alpha/2\), respectively, and, for \(m\geq 1\) (for \(m=0\) this condition is omitted), \(\partial^{2m-p-1}g/\partial t^p \partial x_1^{l_1}\cdots \partial x_{n-1}^{l_{n-1}}\) \((p=0,1,\ldots,m-1)\) satisfies, with respect to \(t\), a Hölder condition with exponent \((1+\alpha)/2\), then we shall say that the surface \(\Gamma\) is of type

\[ \mathcal{L}_{2m-1,1,(1+\alpha)/2}^{m,\alpha_1,\alpha/2} \quad (0<\alpha,\ \alpha_1\leq 1) \]

for \(m=1,2,\ldots\), and of type \(\mathcal{L}^{0,\alpha_1,\alpha/2}\) for \(m=0\).

Introduce the notation:

\[ G(x-\xi,t-\tau)=(t-\tau)^{-n/2} \exp\left\{-\frac14(t-\tau)^{-1}\sum_{i=1}^{n}(x_i-\xi_i)^2\right\}; \]

\(\Gamma_\tau\) is the section of the surface \(\Gamma\) by the hyperplane \(t=\tau\) \((0\leq \tau\leq T<+\infty)\); \(n(x,t)\) is the interior normal to the section \(\Gamma_t\) at the point \((x,t)\in\Gamma_t\); \(Q_T\) is the domain of the space \((x,t)\) bounded by the lateral surface \(\Gamma\) and the hyperplanes \(t=0\) and \(t=T>0\).

Theorem 1. If the surface \(\Gamma\) is of type \(Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}\) \((0<\alpha\leqslant 1)\) and the density \(\Phi(\xi,\tau)\) is a bounded integrable function, then the heat potential of a simple layer

\[ U(x,t)=\int_0^t d\tau \iint_{\Gamma_\tau} G(x-\xi,t-\tau)\Phi(\xi,\tau)\,d\sigma_\xi(\tau) \]

satisfies, in the domain \(\{(x,t), |x_i|<+\infty,\ i=1,2,\ldots,n;\ 0\leqslant t\leqslant T\}\), the Hölder conditions in \(x\) and \(t\) with exponents \(\alpha'\) and \(1/2\), respectively, where \(\alpha'\) is any number for which \(0<\alpha'<1\).

Theorem 2. If the surface \(\Gamma\) is of type \(Л_{1,\alpha,\alpha/2}^{0,1,(1+\alpha)/2}\) and the density \(\Phi(\xi,\tau)\) is a bounded integrable function, then the normal derivative of the heat potential of a simple layer \(V(x,t)=\partial U(x,t)/\partial n(x,t)\) and the heat potential of a double layer

\[ W(x,t)=\int_0^t d\tau \iint_{\Gamma_\tau} \frac{\partial G(x-\xi,t-\tau)}{\partial n(\xi,\tau)} \Phi(\xi,\tau)\,d\sigma_\xi(\tau) \]

satisfy on the surface \(\Gamma\) \(((x,t)\in\Gamma)\) the Hölder conditions in \(x\) and \(t\) with exponents \(\alpha'\) and \(\alpha'/2\), respectively, where \(0<\alpha'<\alpha\), \(\alpha'\) arbitrary.

Remark. If the density \(\Phi(\xi,\tau)\) is continuous on \(\Gamma\), then for \(V(x,t)\) and \(W(x,t)\) the Joukovsky formulas \((^2)\) hold, determining the jumps of these potentials when passing through the surface \(\Gamma\).

Definition. Let \(f(x_1,\ldots,x_k;t)\) be defined in a domain \(B\) and satisfy in this domain one of the conditions A and B of the definition of type \(Л\) for the surface \(\Gamma\) (where \(n-1=k\)). We shall then say that the function \(f(x,t)\) belongs in the domain \(B\) to the class \(H_{2m+1,\alpha_1,\alpha/2}^{m,1,(1+\alpha)/2}(B)\) or \(H_{2m-1,1,(1+\alpha)/2}^{m,\alpha_1,\alpha/2}(B)\), respectively.

Theorem 3. If the surface \(\Gamma\) is of type \(Л_{2m-1,1,(1+\alpha)/2}^{m,\alpha,\alpha/2}\), \(0<\alpha<1\), \(m=1,2,\ldots\) (or \(Л_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}\), \(m=1,2,\ldots\)) and the density \(\Phi(\xi,\tau)\) belongs on the surface \(\Gamma\) to the class \(H_{2(m-1)-1,1,(1+\alpha)/2}^{m-1,\alpha,\alpha/2}(\Gamma)\) (or \(H_{2(m-1)+1,\alpha,\alpha/2}^{m-1,1,(1+\alpha)/2}(\Gamma)\)), and moreover

\[ \frac{\partial^k}{\partial \tau^k}\Phi(\xi,0)=0,\qquad (\xi,0)\in\Gamma,\qquad 0\leqslant k\leqslant m-1, \tag{2} \]

then the potentials \(V(x,t)\) and \(W(x,t)\) belong on the surface \(\Gamma\) to the class \(H_{2(m-1),\alpha',\alpha'/2}^{m-1,1,(1+\alpha)/2}(\Gamma)\) (respectively to the class \(H_{2m-1,1,(1+\alpha)/2}^{m,\alpha',\alpha/2}(\Gamma)\)), where \(0<\alpha'<\alpha\), \(\alpha'\) arbitrary.

Theorem 4. If the surface \(\Gamma\) is of type \(Л_{2m-1,1,(1+\alpha)/2}^{m,\alpha,\alpha/2}\), \(0<\alpha<1\), \(m=1,2,\ldots\) (or \(Л_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}\), \(0<\alpha<1\), \(m=1,2,\ldots\)) and the density \(\Phi(\xi,\tau)\) belongs on the surface \(\Gamma\) to the class \(H_{2(m-1)-1,1,(1+\alpha)/2}^{m-1,\alpha,1/2}(\Gamma)\) (or \(H_{2(m-1)+1,\alpha,\alpha/2}^{m-1,1,(1+\alpha)/2}(\Gamma)\)), and moreover (2) holds, then the heat potential of a simple layer \(U(x,t)\) belongs on the surface \(\Gamma\) to the class \(H_{2(m-1)+1,\alpha,\alpha/2}^{m-1,1,(1+\alpha)/2}(\Gamma)\) (respectively \(H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\Gamma)\)).

Theorem 5. Let the surface \(\Gamma\) be of type \(Л_{2m-1,1,(1+\alpha)/2}^{m,\alpha,\alpha/2}\), \(0<\alpha<1\), \(m=1,2,\ldots\) (or \(Л_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}\)) and let the density \(\Phi(\xi,\tau)\) belong on the surface \(\Gamma\) to the class \(H_{2(m-1),\alpha,\alpha/2}^{m-1,1,(1+\alpha)/2}(\Gamma)\) (or to the class \(H_{2m-1,1,(1+\alpha)/2}^{m,\alpha,\alpha/2}(\Gamma)\)), and moreover (2) holds. Then the heat potential of a simple layer \(U(x,t)\) belongs in the domain \(\overline{Q}_T\) to the class \(H_{2m-1,1,(1+\alpha)/2}^{m,\alpha,\alpha/2}(\overline{Q}_T)\) (respectively \(H_{2m+1,\alpha,\alpha/2}^{m,1,(1+\alpha)/2}(\overline{Q}_T)\)).

The proof of Theorems 1–5 (rather complicated, especially for Theorems 3–5) is carried out by classical methods close to those presented in \((^1)\) for the ordinary theory of potential.

§ 2. Consider the third boundary-value problem for the heat-conduction equation

\[ \sum_{i=1}^{n}\frac{\partial^2 u}{\partial x_i^2}-\frac{\partial u}{\partial t}=f(x,t),\qquad (x,t)\in Q_T; \tag{3} \]

\[ u(x,0)=\psi(x),\qquad x\in \Omega=\overline{Q}_T\cap\{t=0\}; \tag{4} \]

\[ \frac{\partial u(x,t)}{\partial n(x,t)}+b(x,t)u(x,t)=\varphi(x,t),\qquad (x,t)\in\Gamma . \tag{5} \]

Theorem 6. Let the surface \(\Gamma\) be of type \(J^{1,\beta,\beta/2}_{1,1,(1+\beta)/2}\) \((0\leq \alpha<\beta<1,\) where \(\beta\) is arbitrary), let the function \(f(x,t)\) belong to the class \(H^{0,\alpha,\alpha/2}(\overline{Q}_T)\), let the function \(\psi(t)\) have partial derivatives \(\partial^2\psi/\partial x_i\partial x_j\) satisfying in \(x\) in \(\Omega\) a Hölder condition with exponent \(\alpha\), and let the functions \(b(x,t)\) and \(\varphi(x,t)\) belong to the class \(H^{0,1,(1+\alpha)/2}_{1,\alpha,\alpha/2}(\Gamma)\), with the right-hand side \(f\) of (3), the initial function \(\psi\), and the boundary function \(\varphi\) being compatible, in view of equation (3), on the edge \(\Omega\cap\Gamma\). Then there exists a solution \(u(x,t)\) of problem (3)—(5) belonging to the class \(H^{1,\alpha,\alpha/2}_{1,1,(1+\alpha)/2}(\overline{Q}_T)\).

Proof. In view of the conditions imposed on \(f\) and \(\psi\), and of Gevrey’s results \((^2)\) for the heat volume potential and the Poisson integral, instead of problem (3)—(5) one may consider the problem \((3^0),(4^0),(5)\), putting in (3) and (4) \(f(x,t)\equiv \psi(x)\equiv 0\). The solution of problem \((3^0),(4^0),(5)\) is sought in the form of a heat potential of a simple layer with unknown density \(\Phi(\xi,\tau)\). The use of boundary condition (5) leads, in view of the remark to Theorem 2, to a Volterra integral equation of the second kind for determining \(\Phi(\xi,\tau)\). Theorems 1—4 show that \(\Phi(\xi,\tau)\) then possesses all the smoothness properties required in Theorem 5 (with \(m=1\)), the application of which completes the proof of Theorem 6.

§ 3. Consider the third boundary-value problem for the parabolic equation

\[ \sum_{i,j=1}^{n}a_{ij}(x,t)\frac{\partial^2u}{\partial x_i\partial x_j} +\sum_{i=1}^{n}b_i(x,t)\frac{\partial u}{\partial x_i} +c(x,t)u-\frac{\partial u}{\partial t} =f(x,t),\qquad (x,t)\in Q_T \tag{6} \]

with the initial condition (4) and the boundary condition

\[ \frac{\partial u(x,t)}{\partial N(x,t)}+b(x,t)u(x,t)=\varphi(x,t),\qquad (x,t)\in\Gamma, \tag{7} \]

where \(\partial/\partial N(x,t)\) is the derivative along the conormal to the surface \(\Gamma\) at the point \((x,t)\).

The \((2+\alpha)\) a priori estimate for the solution of the third boundary-value problem (6), (4), (7) in the closed domain \(\overline{Q}_T\), obtained by the author jointly with V. N. Maslennikova in \((^3)\), makes it possible, with the aid of Theorem 6, to prove by the classical method of continuation with respect to a parameter the following existence theorem.

Theorem 7. Let the surface \(\Gamma\) be of type \(J^{1,\beta,\beta/2}_{1,1,(1+\beta)/2}\) \((0<\alpha<\beta<1,\) where \(\beta\) is arbitrary), let equation (6) be of parabolic type in \(\overline{Q}_T\), and let the coefficients (6) \(a_{ij}(x,t)\), \(b_i(x,t)\), and \(c(x,t)\) belong to the class \(H^{0,\alpha,\alpha/2}(\overline{Q}_T)\). Let the functions \(f(x,t)\) (in \(\overline{Q}_T\)), \(\psi(x)\) (in \(\Omega\)), \(b(x,t)\), \(\varphi(x,t)\), and also \(a_{ij}(x,t)\) (on \(\Gamma\)) satisfy the conditions of Theorem 6, with \(f(x,t)\), \(\psi(x)\), and \(\varphi(x,t)\) being compatible in view of equation (6). Then there exists a solution \(u(x,t)\) of problem (6), (4), (7) belonging to the class \(H^{1,\alpha,\alpha/2}_{1,1,(1+\alpha)/2}(\overline{Q}_T)\).

Moscow State University
named after M. V. Lomonosov

Received
23 VI 1964

CITED LITERATURE

\(^1\) N. M. Günter, Theory of Potential and Its Application to the Basic Problems of Mathematical Physics, Moscow, 1953.
\(^2\) M. Gevrey, J. Math. pure et appl., 9, No. 1—4, 305 (1913).
\(^3\) L. I. Kamynin, V. N. Maslennikova, DAN, 153, No. 3, 526 (1963).