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Mathematics
V. P. Mikhailov
ON A BOUNDARY-VALUE PROBLEM
(Presented by Academician I. G. Petrovskii on 29 V 1962)
Consider the differential equation
\[ L(u)\equiv (-1)^{[s/2]+1}\frac{\partial^s u}{\partial t^s} +(-1)^{m+1}\sum_{|i|=|j|=m} D^i A^{ij}(x,t)D^j u+B(u)=f(x,t), \tag{1} \]
\[ B(u)=\sum_{|i|/2m+s_1/s<1} B^{(i,s_1)}(x,t)\, \frac{\partial^{s_1}}{\partial t^{s_1}}D^i u, \tag{2} \]
where \(s\geqslant 1,\ m\geqslant 1\) are certain integers; \(x=(x_1,\ldots,x_n)\in\Omega_0\); \(\Omega_0\) is a domain bounded by a sufficiently smooth closed surface \(\gamma\) in \(x\)-space; \(t\in[0,T]\); \((x,t)\in Q\equiv \Omega_0\times(0,T)\); \(\Gamma=\gamma\times[0,T]\). The functions
\(A^{ij}(x,t)=A^{ji}(x,t)\), \(B^{(i,s_1)}(x,t)\) are sufficiently smooth in \(\overline Q\); for simplicity we shall regard them as real-valued; \(f(x,t)\in\mathcal L_2(Q)\);
\[ i=(i_1,\ldots,i_n),\quad j=(j_1,\ldots,j_n);\quad |i|=i_1+\cdots+i_n;\quad D^i=\partial^{|i|}/\partial x_1^{i_1}\cdots \partial x_n^{i_n}, \]
\[ \sum_{|i|=|j|=m}\xi_1^{i_1}\cdots \xi_n^{i_n} A^{ij}(x,t)\xi_1^{j_1}\cdots \xi_n^{j_n} \geqslant \theta^2>0 \tag{3} \]
for \(|\xi|=1,\ (x,t)\in\overline Q\).
We shall be interested in the question of existence and uniqueness of a solution of equation (1) in the domain \(Q\) under the boundary conditions
\[ \left.\frac{\partial^r u}{\partial n_{(x,t)}^r}\right|_{\Gamma} =\varphi_r(x,t),\qquad 0\leqslant r\leqslant m-1; \tag{4} \]
\[ \left.\frac{\partial^r u}{\partial t^r}\right|_{t=0} =\psi_r(x);\qquad 0\leqslant r\leqslant k,\quad \text{if } s=2k+1; \tag{5} \]
\[ 0\leqslant r\leqslant k-1,\quad \text{if } s=2k; \]
\[ \left.\frac{\partial^r u}{\partial t^r}\right|_{t=T} =\chi_r(x),\qquad 0\leqslant r\leqslant k-1, \tag{6} \]
where \(n_{(x,t)}\) is the normal to \(\Gamma\) at the point \((x,t)\);
\[ \varphi_r(x,t)\in W_{t,x,2}^{(2m-r-1/2)/s,\,2m-r-1/2}(\Gamma),\qquad \psi_r(x)\in W_{x,2}^{2m(1-r/s-1/2s)}(\Omega_0), \]
\[ \chi_r(x)\in W_{x,2}^{2m(1-r/s-1/2s)}(\Omega_T),\qquad \Omega_T=\overline Q\cap(t=T). \]
We shall assume, in addition, that the boundary functions satisfy the natural compatibility conditions.
Let us note that in the case of odd \(s=2k+1\), instead of problem (1), (4)—(6) one may consider the problem \((\widetilde{1}),(\widetilde{4}),(\widetilde{5}),(\widetilde{6})\), where \((\widetilde{1})\) differs from (1) by the sign of \(\partial^s u/\partial t^s=\partial^{2k+1}u/\partial t^{2k+1}\), and the conditions \((\widetilde{5})\) and \((\widetilde{6})\) are obtained from (5) and (6) if the latter are interchanged.
Equation (1) for \(s=1\) is parabolic, and the problem considered for it has been well studied. For \(s=3\) the problem (1), (4)—(6) in the case,
when \(B(u)\equiv 0\), was studied by A. A. Dezin; in \((^{1,2})\) he constructed a generalized solution of this problem*.
Theorem 1 on an a priori estimate. Let \(u(x,t)\in W_{t,x,2}^{(p,q)}(Q)\), \(p\ge s,\ q\ge 2m\). Then there exists a constant \(C\), depending only on the coefficients of (1) and on the domain \(Q\), such that
\[ \|u\|_{W^{(p,q)}(Q)}^{2}\le C^{2}\left(\|u\|_{\mathscr L_2(Q)}^{2}+\|Lu\|_{W^{(p-s,q-2m)}(Q)}^{2}+\right. \]
\[ \left. +\sum_{r=0}^{m-1}\|\varphi_r\|_{W^{(p_r,q_r)}(\Gamma)}^{2} +\sum_{r=0}^{k}\|\psi_r\|_{W^{q'_r}(\Omega_0)}^{2} +\sum_{r=0}^{k-1}\|\chi_r\|_{W^{q'_r}(\Omega_T)}^{2} \right), \tag{7} \]
where \(p_r=p(1-r/q-1/2q)\), \(q_r=q(1-r/q-1/2q)\), \(q'_r=q(1-r/p-1/2p)\).
If \(p=s,\ q=2m\), then estimate (7) can be obtained in the following way: introduce the new variable \(w(x,t)=u(x,t)-z(x,t)\), where \(z(x,t)\in W^{(s,2m)}(Q)\) and satisfies the boundary conditions (4)—(6) (such a function \(z(x,t)\) can be constructed according to \((^3)\)); \(w(x,t)\) satisfies the homogeneous conditions (4)—(6) (we shall call them conditions \((4_0)—(6_0)\)) and equation (1) with right-hand side \(F(x,t)=f(x,t)-L(z)\); moreover,
\[ \|F\|_{\mathscr L_2(Q)}^{2}\le C_1^{2}\left(\|f\|_{\mathscr L_2(Q)}^{2}+\|z\|_{W^{s,2m}(Q)}^{2}\right)\le \]
\[ \le C_2^{2}\left( \sum\|\varphi_r\|_{W^{(p_r,q_r)}(\Gamma)}^{2} +\sum\|\psi_r\|_{W^{q'_r}(\Omega_0)}^{2} +\|f\|_{\mathscr L_2(Q)}^{2} +\sum\|\chi_r\|_{W^{q'_r}(\Omega_T)}^{2} \right). \]
The estimate of \(\|w\|_{W^{(s,2m)}(Q)}^{2}\) is obtained by multiplying (1) by \(\partial^s w/\partial t^s\) and integrating the resulting equality over \(Q\).
In the remaining cases \(p\ge s,\ q\ge 2m\), to obtain (7) one should use a partition of unity in \(\overline Q\), the “parametrix” of the fundamental solution for interior points of \(Q\), or the “parametrix” of the Green function in a neighborhood of boundary points. With the aid of the “parametrix” of the Green function one can obtain an a priori estimate also under more general boundary conditions.
Theorem 2 on an a priori estimate. Let \(u(x,t)\in W^{p,q}(Q)\). Then there exists a constant \(C\), independent of \(u(x,t)\), such that
\[ \|u\|_{W^{(p,q)}(Q)}^{2}\le C^{2}\left(\|u\|_{\mathscr L_2(Q)}^{2}+\|L(u)\|_{W^{p-s,q-2m}(Q)}^{2}+\right. \]
\[ \left. +\sum\|A_r(x,t,u)\|_{W^{\widetilde p_r,\widetilde q_r}(\Gamma)}^{2} +\sum\|B_r(x,u)\|_{W^{\widetilde q'_r}(\Omega_0)}^{2} +\sum\|C_r(x,u)\|_{W^{\widetilde q''_r}(\Omega_T)}^{2} \right), \tag{8} \]
where
\[ A_r(x,t,u)=\sum_{j=0}^{l_r}\alpha_{rj}(x,t)\, \frac{\partial^{j}u}{\partial n_{(x,t)}^{j}}, \qquad 0\le r\le m-1; \]
\[ B_r(x,u)=\sum_{j=0}^{m_r}\beta_{rj}(x)\frac{\partial^{j}u}{\partial t^{j}}; \qquad 0\le r\le k \quad \text{for } s=2k+1; \]
\[ 0\le r\le k-1 \quad \text{for } s=2k; \]
\[ C_r(x,u)=\sum_{j=0}^{n_r}\gamma_{rj}(x)\frac{\partial^{j}u}{\partial t^{j}}, \qquad 0\le r\le k-1; \]
* Note added in proof. Recently it has become known to the author that in \((^{10})\) the existence of a generalized solution of problem (1), (4)—(6) was proved under the conditions: a) \(A_{ij}\) are constants; b) \(B(u)\equiv 0\); c) \(s<m\); d) either \(s=4s_1,\ m=2m_1+1\), or \(s=4s_1+2,\ m=2m_1\).
\(\alpha_{ri}, \beta_{ri}, \gamma_{ri}\) are sufficiently smooth functions given on \(\Gamma, \Omega_0, \Omega_T\), respectively; the operators \(A_r, B_r, C_r\) cover, in the sense of (4), the operator \(L\);
\(\tilde p_r = p(1-l_r/q-1/2q)\); \(\tilde q_r = q(1-l_r/q-1/2q)\); \(\tilde{\tilde q}_r = q(1-m_r/p-1/2p)\); \(\tilde{\tilde q}_r = q(1-n_r/p-1/2p)\), \(p \geq \max(s,m_r,n_r)\), \(q \geq \max(2m,l_r)\). In the parabolic case, \(s=1\), Theorem 2 was formulated in [5].
Consider the special case of equation (1)
\[ L_0(u)\equiv (-1)^{[s/2]+1}\frac{\partial^s u}{\partial t^s}-(-\Delta)^m u=f(x,t). \tag{9} \]
Theorem 3. The problem (9), (4)—(6) is uniquely solvable in \(W^{(s,2m)}(Q)\) for arbitrary \(f(x,t)\in \mathcal L_2(Q)\); \(\varphi_r(x,t)\in W^{(2m-r-1/2s)2m,(2m-r-1/2)}(\Gamma)\),
\(\psi_r\in W^{2m(1-r/s-1/2s)}(\Omega_0)\); \(\chi_r\in W^{2m(1-r/s-1/2s)}(\Omega_T)\).
To prove Theorem 3, again by means of the substitution \(u(x,t)=w(x,t)-z(x,t)\), we reduce the problem (9), (4)—(6) to the problem (9), \((4_0)\)—\((6_0)\) with right-hand side in (9) \(F(x,t)=f(x,t)-L_0(z)\). The latter problem is solved as follows: first a generalized solution \(w(x,t)\in \overset{\circ}{W}{}^{[s/2],m}(Q)\) is constructed, satisfying the integral identity
\[ \left[ \frac{\partial^{[s/2]}w}{\partial t^{[s/2]}}, \frac{\partial^{[s/2]+1}v}{\partial t^{[s/2]+1}} \right] - \sum_{|i|=m}[D^i w,D^i v]=[f,v]. \tag{10} \]
The brackets \([\, ,\,]\) denote integration over \(Q\), and the function \(v(x,t)\) is any function from \(\overset{\circ}{W}{}^{[s/2]+1,m}(Q)\) satisfying the boundary conditions \((4_0)\) and
\[ v\bigg|_{\substack{t=0\\ t=T}} =\cdots= \frac{\partial^{k-1}v}{\partial t^{k-1}}\bigg|_{\substack{t=0\\ t=T}} = \frac{\partial^k v}{\partial t^k}\bigg|_{t=T} =0. \]
Next, uniqueness of the generalized solution in \(\overset{\circ}{W}{}^{[s/2],m}(Q)\) is proved, from which one can obtain the estimate
\[ \|w\|_{\overset{\circ}{W}{}^{[s/2],m}(Q)}^2 \leq C_3^2\|f\|_{\mathcal L_2(Q)}^2, \tag{11} \]
where \(C_3\) does not depend on \(f(x,t)\). After this it is proved that the generalized solution is in fact a smooth solution from \(W^{s,2m}(Q)\) of equation (9). The proof of the smoothness of the solution inside the domain \(Q\) and near those portions of its boundary which do not include some neighborhood of the set \((\Gamma\cap(t=0))\cup(\Gamma\cap(t=T))\) is carried out with the aid of a partition of unity and Hörmander’s theorems. (We note that equation (1), as can be proved, is hypoelliptic.)
The proof of smoothness near \(t=0\) (\(t=T\)) (and hence also near the manifold \((\Gamma\cap(t=0))\cup(\Gamma\cap(t=T))\)) is carried out in the following way: in (10), as the function \(v(x,t)\) one substitutes
\[ \rho(t)G_{N,\delta}(x,\xi,t,\tau) = \rho(t)\sum_{r=1}^{N}V_r(x)V_r(\xi) \int_0^\infty M_r(t,t_1)\xi_\delta(t-t_1)\,dt_1, \quad \delta>0,\ N>l; \tag{12} \]
where \(V_r(x)\) is the \(r\)-th eigenfunction of the operator \((-\Delta)^m\) in the domain \(\Omega_0\), corresponding to the eigenvalue \(\lambda_r\), under the boundary conditions \(V_r|_\gamma=\ldots\)
\[ \ldots=D^{m-1}V_r|_\gamma=0; \]
\(M_r(t,t_1)\) is the Green’s function of the operator
\[ (-1)^{[s/2]}\frac{\partial^s}{\partial t^s}-\lambda_r, \]
satisfying the conditions
\[ M_r(t,t_1)\big|_{t=0} =\cdots= \frac{d^{[s/2]-1}}{dt^{[s/2]-1}}M_r(t,t_1)\big|_{t=0} =0,\qquad M_r\big|_{t=\infty}=0, \]
\[ \xi_\delta(t_1-\tau)=\frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{i\theta(t_1-\tau)}e^{-\delta\theta^2}\,d\theta, \qquad \delta>0; \]
\(\rho(t)\) is a function from \(C^\infty([0,T])\); \(|\rho(t)| \leqslant 1\); \(\rho(t)=1\) for \(0 \leqslant t \leqslant T/3\); \(\rho(t)=0\) for \(T/2 \leqslant t \leqslant T\).
From equality (10) we obtain
\[ w_{N\delta}(\xi,\tau) = [tp,G_{N\delta}] + \sum_{p\leqslant k} B_p\bigl[(u\rho^{2k+1-p})_{tp},G_{N\delta}\bigr] + \sum_{p\geqslant k+1} B_p \left[ (u\rho^{2k+1-p})_{t^k} \frac{\partial^{p-k}G_{N\delta}}{\partial t^{p-k}} \right], \tag{13} \]
where \(w_{N\delta}(\xi,\tau)=[w(x,t),L_0^*(x,t)G_{N\delta}(x,\xi,t,\tau)]\) are smooth functions, and \(B_p,\ p=0,\ldots,2k,\) are certain numbers.
It can be proved that for any \(w(x,t)\in \mathscr L_2(Q)\) the sequence \(w_{N\delta}(\xi,\tau)\) as \(N\to\infty\) and \(\delta\to0\) converges in \(\mathscr L_2(Q)\) to \(w(\xi,\tau)\).
Using (13) and (11), one can prove
\[ \left\| \frac{\partial^{[s/2]+1} w_{N\delta}}{\partial \tau^{[s/2]+1}} \right\|_{\mathscr L_2(Q)}^2 \leqslant C_4^2\|f\|_{\mathscr L_2(Q)}^2, \]
whence, in turn, follows the existence of the \(([s/2]+1)\)-st derivative with respect to \(\tau\) in \(\mathscr L_2(Q\cap(0\leqslant t\leqslant T/3))\) of \(w(\xi,\tau)\). Analogously, by means of a suitably chosen \(\widetilde G_{N\delta}(x,\xi,t,\tau)\), the existence of the \(([s/2]+1)\)-st derivative in \(\mathscr L_2(Q\cap(2T/3\leqslant t\leqslant T))\) is proved, and hence, by virtue of (6), also in \(\mathscr L_2(Q)\). Then the existence in \(\mathscr L_2(Q)\) of the \(([s/2]+2)\)-nd derivative of \(w(\xi,\tau)\) with respect to \(\tau\), etc., up to the derivative with respect to \(\tau\) of order \(s\) of \(w(\xi,\tau)\), is established. In this case the estimate
\[ \|w_{t^s}\|_{\mathscr L_2(Q)}^2 \leqslant C_5^2\|f\|_{\mathscr L_2(Q)}^2 \]
holds.
From this estimate, equation (9) for \(w(x,t)\), and a known theorem from (8), the proof of Theorem 3 follows.
By the method of continuation with respect to a parameter (for the parabolic equation (1) (\(s=1\)) this method is presented in (9)), with the aid of Theorem 1 on an a priori estimate, the following theorems are proved.
Theorem 4. For problem (1), (4)—(6) and its adjoint problem \((1^*)\), \((4^*)\)—\((6^*)\), all Fredholm theorems are valid.
In particular, for the existence of a unique solution of (1), (4)—(6) it is necessary and sufficient that problem \((1_0)\), \((4_0)\)—\((6_0)\) have only the trivial solution.
Theorem 5. If for every smooth finite function \(v(x,t)\) in \(Q\) the inequality
\[ \bigl[ (-1)^m\sum D^i A^{ij}(x,t)D^jv - B(v),\, v \bigr] \geqslant C_6^2[v,v] \]
holds with a constant \(C_6\) independent of \(v\), then problem (1), (4)—(6) is uniquely solvable in \(W^{(s,2m)}(Q)\) for all right-hand sides and boundary functions under consideration.
Theorem 6. Problem (1), (4)—(6) is uniquely solvable in \(W^{s,2m}(Q)\), if the height \(T\) of the cylinder \(Q\) is sufficiently small.
Received
22 V 1962
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