MATHEMATICS

L. I. KAMYNIN, V. N. MASLENNIKOVA

BOUNDARY ESTIMATES FOR THE SOLUTION OF THE THIRD BOUNDARY-VALUE PROBLEM FOR A PARABOLIC EQUATION

(Presented by Academician S. L. Sobolev, 23 VI 1963)

For elliptic equations, \((2+\alpha)\) a priori estimates up to the boundary were obtained by Schauder \((^1,^2)\) for the solution of the first boundary-value problem, by Fiorenza \((^3)\) for the boundary-value problem with oblique derivative, and by Agmon, Douglis, and Nirenberg \((^4)\) for boundary-value problems with general boundary conditions. In the theory of boundary-value problems for parabolic equations, up to the present time a priori estimates (up to the boundary) have been obtained only for solutions of the first boundary-value problem (see \((^5,^6)\)). In our note a \((2+\alpha)\) a priori estimate (up to the boundary) is established for the solution of the third boundary-value problem (with conormal derivative) for a parabolic equation of the second order and for certain parabolic systems.

§ 1. We consider the solution \(u(x,t)\) of the following boundary-value problem:

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

\[ u(x,0)=\psi(x),\qquad x\in \Omega=\overline{Q}\cap\{t=0\}; \tag{2} \]

\[ \frac{\partial u(x,t)}{\partial N}+b(x,t)u(x,t)=\varphi(x,t),\qquad (x,t)\in \Gamma, \tag{3} \]

where \(\partial/\partial N\) is the derivative along the conormal to the surface \(\Gamma\) at the point \((x,t)\). \(Q\) is a domain (possibly unbounded in the \(x_i\)) in the space \((x,t)=(x_1,\ldots,x_n,t)\), lying between the hyperplanes \(t=0\) and \(t=T>0\) and bounded by the lateral surface \(\Gamma\). By \(\partial Q\) we denote the “normal” boundary of \(Q\), i.e. the set \(\partial Q=\Gamma\cup\Omega\).

\(A_1\). The lateral surface \(\Gamma\) has at each of its points a tangent plane nowhere orthogonal to the axis \(Ot\). Moreover, for each point \(P(x,t)\in\Gamma\) there exists an \((n+1)\)-dimensional sphere \(S_\delta(P)\) with center at \(P\) and radius \(\delta>0\) (where \(\delta\) does not depend on the choice of the point \(P\) on \(\Gamma\)) such that the part of \(\Gamma\) lying in the sphere \(S_\delta(P)\) can be represented, for some \(i\) \((1\le i\le n)\), in the form

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

where the function \(h\) has derivatives with respect to \(x_k\) \((k=1,\ldots,i-1,i+1,\ldots,n)\) up to the second order inclusive, satisfying the Hölder condition in \(x\) and \(t\) with exponent \(\alpha>0\), and a first derivative with respect to \(t\), satisfying the Hölder condition in \(x\) and \(t\) with exponent \(\alpha>0\).

For what follows it is convenient to introduce the following norms. Let

\[ d(P_1,P_2)= \left( \sum_{i=1}^{n} (x_i^{(1)}-x_i^{(2)})^2 + |t_1-t_2| \right)^{1/2} \]

be the parabolic distance between the points \(P_1(x^{(1)},t_1)\) and \(P_2(x^{(2)},t_2)\). For

For a function \(u(x,t)=u(P)\) defined on the set \(B\), put

\[ [u]^B_0=\sup_B |u(P)|,\qquad [u]^B_1=\sup_{i,P}\left|\frac{\partial u(P)}{\partial x_i}\right|, \]

\[ [u]^B_2=\max\left\{ \sup_{i,j,P}\left|\frac{\partial^2 u(P)}{\partial x_i\partial x_j}\right|, \sup_P\left|\frac{\partial u(P)}{\partial t}\right| \right\}, \]

\[ [u]^B_\alpha=\sup_{P_1,P_2}\bigl(d^{-\alpha}(P_1,P_2)|u(P_1)-u(P_2)|\bigr), \]

\[ [u]^B_{1+\alpha}=\max\left\{ \sup_{i,P_1,P_2} \left( d^{-\alpha}(P_1,P_2) \left| \frac{\partial u(P_1)}{\partial x_i} - \frac{\partial u(P_2)}{\partial x_i} \right| \right), \right. \]

\[ \left. \sup_{P,P_0} \left( |t_1-t_2|^{-\frac{1+\alpha}{2}} |u(P)-u(P_0)| \right) \right\}, \]

where \(P=P(x,t_1)\), \(P_0=P_0(x,t_2)\),

\[ [u]^B_{2+\alpha}=\max\left\{ \sup_{i,j,P_1,P_2} \left( d^{-\alpha}(P_1,P_2) \left| \frac{\partial^2 u(P_1)}{\partial x_i\partial x_j} - \frac{\partial^2 u(P_2)}{\partial x_i\partial x_j} \right| \right), \right. \]

\[ \sup_{P_1,P_2} \left( d^{-\alpha}(P_1,P_2) \left| \frac{\partial u(P_1)}{\partial t} - \frac{\partial u(P_2)}{\partial t} \right| \right), \]

\[ \left. \sup_{i,P,P_0} \left( |t_1-t_2|^{-\frac{1+\alpha}{2}} \left| \frac{\partial u(P)}{\partial x_i} - \frac{\partial u(P_0)}{\partial x_i} \right| \right) \right\}, \]

\[ |u|^B_\alpha=[u]^B_0+[u]^B_\alpha,\qquad |u|^B_1=[u]^B_0+[u]^B_1,\qquad |u|^B_2=|u|^B_1+[u]^B_2, \]

\[ |u|^B_{1+\alpha}=|u|^B_1+[u]^B_{1+\alpha},\qquad |u|^B_{2+\alpha}=|u|^B_2+[u]^B_{2+\alpha}. \]

Concerning the coefficients of equation (1) and the functions entering into (2) and (3), we shall assume that the following conditions are satisfied:

\(A_2.\) Equation (1) is of parabolic type in \(\overline Q\); the matrix \(a_{ij}(x,t)\) is symmetric with a positive-definite quadratic form in \(\overline Q\), and
\[ \det |a_{ij}(x,t)|\ge a>0. \]

\[ A_3.\quad |a_{ij}|^{\overline Q}_\alpha+|b_i|^{\overline Q}_\alpha+|c|^{\overline Q}_\alpha\le M_1,\qquad |f|^{\overline Q}_\alpha<+\infty. \]

\[ A_4.\quad |b|^\Gamma_{1+\alpha}\le M_2,\qquad |\varphi|^\Gamma_{1+\alpha}<+\infty,\qquad |\psi|^\Omega_{2+\alpha}<+\infty. \]

\(A_5.\) The right-hand side \(f\) from (1), the initial function \(\psi\) from (2), and the boundary function \(\varphi\) from (3) are assumed to be compatible in virtue of equation (1), i.e. they satisfy equation (1) on \(\Omega\cap\Gamma\).

§ 2. Lemma 1 (cf. (1, \({}^6\))). If \(u(x,t)\) is a solution of the homogeneous parabolic equation with constant coefficients

\[ L^0u=\sum_{i,j=1}^n a^0_{ij}\frac{\partial^2 u}{\partial x_i\partial x_j} - \frac{\partial u}{\partial t}=0 \]

(where \(\det |a^0_{ij}|\ge a>0,\ \sup |a^0_{ij}|\le M_1\)) in the rectangle
\[ V_R=\{(x,t),\ |x_i|<R;\ i=1,2,\ldots,n;\ 0<t<R^2\}, \]
continuous in \(\overline V_R\), then

\[ [u]^{V_{R,\lambda}}_{2+\alpha} \le C(a,M_1)\lambda^{-(2+\alpha)} \sup_{\partial V_R}|u(P)|, \]

where

\[ V_{R,\lambda}=\{(x,t),\ |x_i|\le R-\lambda,\ i=1,2,\ldots,n;\ \lambda^2\le t\le R^2\},\qquad 0<\lambda\le 1. \]

Lemma 2 (basic). If \(f(x,t)\) is defined in the quarter-space

\[ D=\{(x,t),\ |x_i|<+\infty,\ i=1,2,\ldots,n-1,\ 0<x_n<+\infty,\ 0<t<+\infty\}, \]

with \(|f|_{\alpha}^{D}<+\infty\), the function \(\psi(x)\) is defined in
\(\Omega_1=\overline D\cap\{t=0\}\), with \(|\psi|_{2+\alpha}^{\Omega_1}<+\infty\), and
\(\varphi(x_1,\ldots,x_{n-1};t)\) is defined on the hyperplane

\[ \Gamma_1=\overline D\cap\{x_n=0\}, \]

with \(|\varphi|_{1+\alpha}^{\Gamma_1}<+\infty\), then there exists a function \(v(x,t)\) such that

\[ L^0v=f(x,t),\qquad (x,t)\in D; \tag{4} \]

\[ v(x,0)=\psi(x),\qquad x\in\Omega_1, \tag{5} \]

\[ \frac{\partial v(x,t)}{\partial N_0}=\varphi(x,t),\qquad (x,t)\in\Gamma_1 \tag{6} \]

(where \(\partial/\partial N_0\) is the derivative with respect to the conormal for equation (4) to the hyperplane \(\Gamma_1\) at the point \((x,t)\)), and

\[ [v]_{2+\alpha}^{D}\leq C(a,M_1)\bigl(|f|_{\alpha}^{D}+[\psi]_{2+\alpha}^{\Omega_1}+[\varphi]_{1+\alpha}^{\Gamma_1}\bigr); \tag{7} \]

\[ [v]_{0}^{D_T}\leq C(a,M_1,T)\bigl(|f|_{0}^{D}+[\psi]_{0}^{\Omega_1}+[\varphi]_{0}^{\Gamma_1}\bigr) \tag{8} \]

\[ (D_T=D\cap\{0\leq t\leq T\}). \]

Remark. The right-hand side \(f\) in (4), the initial function \(\psi\) in (5), and the boundary function \(\varphi\) in (6) are, of course, assumed to be compatible by virtue of equation (4).

Proof. The use of the apparatus of heat potentials (cf. \((^7)\)) makes it possible to write the solution \(v(x,t)\) of problem (4)—(6) in explicit form; after this, a sufficiently painstaking investigation of the properties of the heat potentials obtained makes it possible to prove the validity of the estimates (7), (8).

Lemma 3 (cf. \((^{5,6})\)) (on the interior \((2+\alpha)\) a priori estimate). Let conditions \(A_1—A_5\) be satisfied with respect to the domain \(Q\) and the data of problem (1)—(3). Let problem (1)—(3) have a solution \(u(x,t)\), for which \(|u|_{2+\alpha}^{\overline Q}<+\infty\). Then there exists a constant \(C(Q,a,M_1)\) such that, for an interior domain \(Q_1\subset\overline Q\), for which \(d(Q,Q_1)\geq\lambda>0\), the estimate

\[ |u|_{2+\alpha}^{Q_1}\leq C(Q,a,M_1)\bigl(|f|_{\alpha}^{Q}+\lambda^{-(2+\alpha)}[u]_{0}^{Q}\bigr) \]

holds.

Lemma 4 (cf. \((^{5,6})\)) (on the boundary \((2+\alpha)\) a priori estimate near \(\Omega\)). Let the conditions of Lemma 3 be satisfied. Let \(G\subseteq Q\) be a domain for which

\[ G\cap\Omega=\Omega_G\subset\Omega \]

is an interior subset of the domain \(\Omega\) \((\overline{\Omega}_G\cap\Gamma=0)\). Then, for any subdomain \(G_1\subset G\) contained in \(G\), there exists a constant \(C(Q,\Omega_G,a,M_1)\) such that

\[ |u|_{2+\alpha}^{G_1}\leq C(Q,\Omega_G,a,M_1)\bigl(|f|_{\alpha}^{Q}+|\psi|_{2+\alpha}^{\Omega_G}+[u]_{0}^{Q}\bigr). \]

§ 3. With the aid of Lemmas 1—4, using the classical method of Schauder (see \((^{1,2})\), as well as \((^4)\)), the following boundary a priori estimates of the solution of problem (1)—(3) are established.

Theorem 1. Let conditions \(A_1—A_5\) be satisfied. Let \(Q'\subseteq Q\) be a subdomain of \(Q\) (in particular, it may coincide with \(Q\)), with

\[ \Omega'=\overline{Q'}\cap\Omega,\qquad \Gamma'=\overline{Q'}\cap\Gamma. \]

Let \(u(x,t)\) be a solution, bounded in \(\overline Q\), of problem (1)—(3), in which \(\Omega\) in (2) and \(\Gamma\) in (3) are replaced respectively by \(\Omega'\) and \(\Gamma'\). Finally, let

\[ |u|_{2+\alpha}^{\overline Q}<+\infty. \]

Then

\[ |u|_{2+\alpha}^{Q'}\leq C(Q',Q,a,M_1,M_2,\delta) \bigl(|f|_{\alpha}^{Q}+|\varphi|_{1+\alpha}^{\Gamma'}+|\psi|_{2+\alpha}^{\Omega'}+[u]_{0}^{Q}\bigr). \]

Theorem 2. In the case when the domain \(Q\) is bounded and the solution \(u(x,t)\) of problem (1)—(3) \(\bigl(|u|_{2+\alpha}^{Q}<+\infty\bigr)\) is unique, one has

\[ |u|_{2+\alpha}^{Q}\leq C\bigl(|f|_{\alpha}^{Q}+|\varphi|_{1+\alpha}^{\Gamma}+|\psi|_{2+\alpha}^{\Omega}\bigr), \]

where the constant \(C\) depends on the equation, but does not depend on the function \(u(x,t)\).

§ 4. We now consider the following boundary-value problem for a parabolic system of special form:

\[ \sum_{i,j=1}^{n} a_{ij}^{(k)}(x,t)\frac{\partial^{2}u_k}{\partial x_i\partial x_j} +\sum_{l=1}^{m}\sum_{i=1}^{n} b_{il}^{(k)}(x,t)\frac{\partial u_l}{\partial x_i} +\sum_{l=1}^{m} c_l^{(k)}(x,t)u_l -\frac{\partial u_k}{\partial t} =f_k(x,t), \tag{9} \]

\[ k=1,2,\ldots,m,\qquad (x,t)\in Q; \]

\[ u_k(x,0)=\psi_k(x),\qquad x\in\Omega; \tag{10} \]

\[ \frac{\partial u_k(x,t)}{\partial N_k} +\sum_{l=1}^{m} b_l^{(k)}(x,t)u_l(x,t) =\varphi_k(x,t),\qquad (x,t)\in\Gamma, \tag{11} \]

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

Theorem 3. Suppose that for the system (9)—(11), the domain \(Q\), and the surface \(\Gamma\), all conditions \(A_1—A_5\) are satisfied. Let a subdomain \(Q'\Subset Q\) of the domain \(Q\) satisfy the conditions of Theorem 1. Let \(u_k(x,t)\) \((k=1,2,\ldots,m)\) be a solution of problem (9)—(11), bounded in the domain \(\overline Q\), and such that

\[ |u_k|_{2+\alpha}^{\overline Q}<+\infty,\qquad k=1,2,\ldots,m. \]

Then

\[ |u_k|_{2+\alpha}^{Q'} < C(Q';\,Q,\alpha,M_1,M_2,\delta)\max_k \bigl(|f_k|_{\alpha}^{Q}+|\varphi_k|_{1+\alpha}^{\Gamma}+|\psi_k|_{2+\alpha}^{\Omega}+[u_k]_0^{Q}\bigr). \]

§ 5. With the aid of the established boundary a priori estimate, the existence theorems for a solution of problem (1)—(3) or (9)—(11) are proved in the usual way.

Moscow State University
named after M. V. Lomonosov

Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
17 VI 1963

References Cited

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