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MATHEMATICS
Rudolf VYBORNY
ON THE PROPERTIES OF SOLUTIONS OF CERTAIN BOUNDARY-VALUE PROBLEMS FOR EQUATIONS OF PARABOLIC TYPE
(Presented by Academician S. L. Sobolev on 18 VI 1957)
In the present paper we prove the uniqueness of certain boundary-value problems for partial differential equations of parabolic type and the continuous dependence of solutions on the coefficients of the equation, on the right-hand sides, and on the initial and boundary conditions. The method developed by Hopf ($^{1,2}$) and O. A. Oleinik ($^3$) is used.
Let $G$ be a domain in the $n$-dimensional Euclidean space $E_n$, the points of which we denote by $X = (x_1, \ldots, x_n)$; for brevity we shall write
$|X|^2 = x_1^2 + \ldots + x_n^2$, $f(X) = f(x_1, \ldots, x_n)$, $u(X,t)=u(x_1,\ldots,x_n,t)$, etc. Denote the cylinder $G \times (0,T)$ by the symbol $V$, its lower base by the symbol $Z$, and its lateral surface $(0<t<T)$ by the symbol $S$. Put $V_t = G \times (0,t)$ for $0<t<T$. Let
\[ L(u)=\sum_{i,j=1}^{n} a_{ij}u_{x_i x_j}+\sum_{i=1}^{n} b_i u_{x_i}-u_t+cu \]
be a linear differential operator whose coefficients are continuous and bounded functions on $V+S+Z$, with $c \leq 0$, and for all $\lambda_i$ and everywhere in $V+S+Z$ the inequality
\[ \sum_{i,j=1}^{n} a_{ij}\lambda_i\lambda_j \geq m\sum_{i=1}^{n}\lambda_i^2,\quad m=\mathrm{const}>0. \]
Concerning the boundary of the domain $G$ we shall assume that it has the following property: for every point $P \in S$ there exists a sphere $K_P$ such that $K_P \subset V$ and $\overline{K}_P \cap S = \{P\}$. If the boundary of the domain $G$ is formed by a twice continuously differentiable surface, then this condition is obviously satisfied. By $l$ we denote the direction which forms an acute angle with the inward normal of the sphere $K_P$ at the point $P$.
Theorem 1. Let $u(X,t)$ be, in the domain $V$, a solution of the equation
\[ L(u)=f(X,t), \tag{1} \]
continuous on $V+S+Z$. Let $(X_1,t_1)\in S$; $u(X,t)\geq u(X_1,t_1)$ on $V+S+Z$. If $f\leq 0$, $u(X_1,t_1)<0$, and if $u$ is not constant in $V_{t_1}$, then
\[ \lim_{(X,t)\to (X_1,t_1)} -\frac{u(X,t)-u(X_1,t_1)} {\sqrt{|X-X_1|^2+(t-t_1)^2}}>0, \tag{2} \]
where $(X,t)$ varies in the direction $l$.
Remark 1. From Theorem 1 it follows immediately that a nonconstant (in the domain $V_{t_0}$) solution $u$ of the equation
\[ L(u)=0 \tag{3} \]
cannot attain either a positive maximum or a negative minimum at the point \((X_0,t_0)\in S\), if \(\partial u(X_0,t_0)/\partial l=0\).
Remark 2. If \(c\equiv 0\), then \(u\) cannot attain either a maximum or a minimum at the point \((X_0,t_0)\). This assertion contains a generalization of some theorems proved by Adler and Friedman \((^{4-6})\).
We shall preface the proof of Theorem 1 with the following lemma.
Lemma. Let \(K\) be a sphere of radius \(R\), \(K\subset V\). Suppose that at the point \((X_1,t_1)\), lying on its boundary, \(u\) assumes its least negative value. If the point \((X_1,t_1)\) is not a pole* of the sphere \(K\) and if
\[ u(X,t)>u(X_1,t_1) \tag{4} \]
in \(\overline K\) for \((X,t)\ne (X_1,t_1)\), then inequality (2) is valid.
Proof. Place the origin of coordinates at the center of the sphere \(K\). Construct a sphere \(K_1\) with center at the point \((X_1,t_1)\) so that the function \(g(X)=|X|^2\) attains a positive minimum on the set \(\overline K\cap \overline K_1\). For the function \(h=e^{-\alpha(|X|^2+t^2)}-e^{-\alpha R^2}\) we shall have \(L(h)>0\), if we take \(\alpha\) sufficiently large. Therefore the function \(u-\varepsilon h\) cannot attain a minimum on \(K\cap K_1\); consequently, for sufficiently small \(\varepsilon\) we shall have \(u-\varepsilon h\ge u(X_1,t_1)\) in \(\overline K\cap \overline K_1\), whence the assertion of the lemma is already easily obtained.
Proof of Theorem 1. For every point \((X_2,t_2)\in K_P\) for which \(u(X_2,t_2)=u(X_1,t_1)\), it must be, by Nirenberg’s theorem \((^7)\), that \(t_2<t_1\); thus, one may assume that all points of \(K_P\) satisfy inequality (4). To finish the proof, it is now enough to use the preceding lemma.
Theorem 2. Let \(u\) be a solution of equation (3), continuous on \(V+S+Z\), satisfying the boundary condition
\[ l(u)=a\frac{\partial u}{\partial l}+bu=0 \tag{5} \]
on \(S\) and the initial condition
\[ u=0 \tag{6} \]
on \(Z\). Suppose further,** \(a^2(P)+b^2(P)\ge k>0,\ a\ge 0,\ b\le 0\). Then \(u\equiv 0\) in the domain \(V\).
For the proof of Theorem 2 it is enough to use Theorem 1. Let us make one more assumption on the surface \(S\), namely, that it has a continuously varying tangent plane. Then the theorems below can be proved by the same methods as those used by O. A. Oleinik in the work \((^3)\).
Theorem 3. Let \(u(P)\) be a solution of equation (1), continuous on \(V+S+Z\), satisfying conditions (5) and (6). Then, if we denote
\[ M=\sup_{V+S+Z}|f|, \]
then
\[ |u(P)|\le \frac{M}{\inf |c|}. \]
Theorem 4. Suppose that for any functions \(\psi_1(P),\psi_2(P)\) of class \(C_k\) \((k\ge 0)\) there exists a solution of the equation \(L(u)=0\), satisfying the conditions \(u=\psi_1\) on \(Z\), \(l(u)=\psi_2\) on \(S\). If the functions \(a,b\) belong to the class \(C_k\), then the solution of equation (1), satisfying the conditions \(u=\varphi_1\) on \(Z\) and \(l(u)=\varphi_2\) on \(S\), satisfies the inequality
\[ |u(P)|\le \frac{M}{\inf |c|}+m_1+\frac{m_2}{\inf(|a|+|b|)} \left[ K_1+\frac{K_2}{\inf |c|}\sup R \right], \]
where
\[ R=\sum_{i,k=1}^{n}|a_{ik}|+\sum_{i=1}^{n}|b_i|+|c|+1,\qquad M=\sup|f|,\qquad m_1=\sup|\varphi_1|,\qquad m_2=\sup|\varphi_2|. \]
* We call a point of the sphere \(K\) a pole if its ordinate \(t\) is the smallest or the largest among the ordinates \(t\) of all points of the sphere \(K\).
** Everywhere below these assumptions remain in force.
Theorem 5. Let us leave unchanged the notation and assumptions of Theorem 4. Further, let \(\sup c < 0\). Let \(\bar u\) be a solution of the equation
\[ L_1(\bar u)=\sum_{i,j=1}^{n}\bar a_{ij}\bar u_{x_i x_j} +\sum_{i=1}^{n}\bar b_i \bar u_{x_i} -\bar u_t+\bar c\,\bar u, \]
satisfying the conditions \(\bar u=\varphi_1\) on \(Z\) and \(\bar a\,\dfrac{\partial \bar u}{\partial l}+\bar b\bar u=\varphi_2\) on \(S\), and having two continuous and bounded derivatives on \(V+S+Z\). Then the difference \(|u-\bar u|\) will be arbitrarily small provided that the differences of the coefficients of the equations, the differences of the initial and boundary conditions, and the differences of the right-hand sides of the equations are sufficiently small in absolute value.
Theorem 6. Let, for a function \(W\) of class \(C_2\) on \(\bar V\), for which the inequality \(\min_{\bar V} L(W)>0\) holds, there exist a solution \(u_W\) of the equation \(L(u_W)=L(W)\), satisfying homogeneous initial and boundary conditions, and let, further, there exist a solution \(u_2\) of equation (3) such that \(u_2=\varphi_1\) on \(Z\) and \(l(u_2)=\varphi_2\) on \(S\). Then for the solution \(u\) of equation (1), for which the same initial and boundary conditions are fulfilled as for the function \(u_2\), we have the estimate
\[ |u(P)|\leq m_1+\frac{m_2}{\inf |b|}+CM, \]
where \(C\) is a constant depending on the coefficient of the equation and on the functions \(a,b\), but depending neither on \(\varphi_i\) nor on \(f\).
Theorems analogous to Theorems 1–6 can also be proved in the case of a smooth moving boundary \(S\), whose tangent plane is at no point perpendicular to the \(t\)-axis.
Mathematical Institute
Czechoslovak Academy of Sciences
Received
17 V 1957
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