Full Text
V. K. ROMANKO
BOUNDARY-VALUE PROBLEMS FOR OPERATORS IN TWO VARIABLES WITH NONHOMOGENEOUS PRINCIPAL PART
(Presented by Academician L. S. Pontryagin, 16 V 1964)
For operators in two variables \(t, x\), considered in a bounded convex domain with piecewise smooth boundary, boundary-value problems have been well studied in the case of a homogeneous principal part \((^8)\) (in particular, for hyperbolic \((^7)\) and elliptic equations), and also for certain special classes of operators that are, in a certain sense, the most direct generalization of the parabolic and elliptic cases \((^{2,3,5})\). In the cited works, boundary-value problems more general than those considered by us were studied.
In the present work we consider the simplest boundary-value problems for operators with nonhomogeneous principal part. Restricting ourselves for the time being to the case where the principal part is a binomial, we show that, depending on the location of the roots of the characteristic equation, the operators under consideration split into five types. For each type its own class of boundary-value problems is indicated. The method of investigation applied makes it possible to show that these classes of problems describe, in a certain sense, all admissible boundary conditions for each type of operators. This is established by constructing examples showing the ill-posedness or instability of problems different from those considered. The approach used is close to \((^1)\).
We pass to the statement of the results.
Consider, in the domain \(V=(0\leq t\leq T,\ 0\leq x\leq 2l)\), the equation
\[ Lu+L_1(t,x)u=f, \tag{1} \]
where
\[ Lu\equiv \frac{\partial^m u}{\partial t^m}+\frac{\partial^n u}{\partial x^n}; \qquad L_1(t,x)u\equiv \sum_{r=r_1/m+r_2/n_0<1} a_r(t,x)\frac{\partial^r u}{\partial t^{r_1}\partial x^{r_2}}; \]
\(n_0\) is some positive integer, \(n_0<n\), determined by the type of the operator \(L\); \(a_{m-1}(t,x)\in C_{t,x}^{m-1,n}\) and is periodic in \(x\) with period \(2l\); \(a_r(t,x)\), \(r_1\neq m-1\), are summable functions bounded on \(V\), \(f\in \mathcal L_2(V)\).
We shall be interested in the character of the solvability of equation (1) under boundary conditions of the form
\[ u_{t_i}^{(r_i)}(T,x)=0,\qquad i=1,2,\ldots,\nu; \]
\[ u_{t_j}^{(s_j)}(0,x)=0,\qquad j=1,2,\ldots,m-\nu, \tag{\(\Gamma\)} \]
where \(r_i, s_j\) are integers and \(0\leq r_1<r_2<\cdots<r_\nu<m,\ 0\leq s_1<s_2<\cdots<s_{m-\nu}<m\), and under periodicity conditions in \(x\). Here \(\nu\) is a positive integer that determines, for each type of operator \(L\), its own class of problems.
Consider the equation determined by the operator \(L\),
\[ \omega^m+(i\xi)^n=0. \tag{2} \]
For real \(\xi\neq 0\), among the roots \(\omega_j,\ i=1,2,\ldots,m\), of equation (2) there are no multiple roots. The possible types of location of the roots \(\omega_j\) do not depend on the values \(\xi\neq 0\) and determine the following five distinct types of the operator \(L\):
I. \(m=2m_1\), and equation (2) has \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j>0\) and \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j<0\). Then \(\nu=m_1\) (Dirichlet problem).
II. \(m=2m_1\) and equation (2) has \((m_1-1)\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j>0\) and \((m_1-1)\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j<0\). Moreover, two roots are equal to \(\pm i\). Then \(\nu=m_1-1\) or \(\nu=m_1+1\).
III. \(m=2m_1+1\) and equation (2) has \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j>0\), \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j<0\), and one purely imaginary root. Then \(\nu=m_1\) or \(\nu=m_1+1\).
IV. \(m=2m_1+1\) and equation (2) has \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j>0\) and \((m_1+1)\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j<0\). Then \(\nu=m_1\).
V. \(m=2m_1+1\) and equation (2) has \((m_1+1)\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j>0\) and \(m_1\) roots \(\omega_j\) with \(\operatorname{Re}\omega_j<0\). Then \(\nu=m_1+1\).
In what follows we shall speak of the operator \(L\) and of the boundary conditions \((\Gamma)\), understanding by \(L\) any one of the listed types I–V and associating with each type its own number of boundary conditions \(\nu\) in \((\Gamma)\), defined above.
Denote by \(P_{t,x}^{m+1,n+1}\) the set of complex functions having continuous periodic partial derivatives of period \(2l\) in \(t\) up to order \(m+1\) and in \(x\) up to order \(n+1\). Let, for \(u\in P_{t,x}^{m+1,n+1}\),
\[ u=\sum_{k=-\infty}^{+\infty} u_k(t)e^{ik\pi x/l},\qquad Lu\equiv \varphi=\sum_{k=-\infty}^{+\infty}\varphi_k(t)e^{ik\pi x/l}. \]
If \(u\in P_{t,x}^{m+1,n+1}\) satisfies \((\Gamma)\) and the equation
\[ \frac{\partial^m u}{\partial t^m}+\frac{\partial^n u}{\partial x^n}=\varphi, \tag{3} \]
then \(u_k(t)\) satisfies the conditions
\[ \frac{d^m u_k}{dt^m}+\left(\frac{ik\pi}{l}\right)^n u_k=\varphi_k; \tag{4} \]
\[ u_k^{(r_i)}(T)=0,\quad i=1,2,\ldots,\nu;\qquad u_k^{(s_j)}(0)=0,\quad j=1,2,\ldots,m-\nu, \tag{5} \]
\[ 0\le r_1<r_2<\cdots<r_\nu<m;\quad 0\le s_1<s_2<\cdots<s_{m-\nu}<m. \]
Lemma 1. For each of the operators of types I–V, for all \(k\) the following inequalities hold:
\[ \text{I.}\qquad |k|^r|u_k(t)|\le C_1 |k|^{-(n-n/4m_1-r)}\|\varphi_k\|_t,\qquad r=0,1,\ldots,n_0=\left[n-\frac{n}{4m_1}\right]. \]
\[ \text{II.}\qquad |k|^r|u_k(t)|\le C_2 |k|^{-(n-n/2m_1-r)}\|\varphi_k\|_t,\qquad r=0,1,\ldots,n_0=\left[n-\frac{n}{2m_1}\right]. \]
\[ \text{III.}\qquad |k|^r|u_k(t)|\le C_3 |k|^{-(n-n/(2m_1+1)-r)}\|\varphi_k\|_t,\qquad r=0,1,\ldots,n_0=\left[n-\frac{n}{2m_1+1}\right]. \]
\[ \text{IV.}\qquad |k|^r|u_k(t)|\le C_4 |k|^{-(n-n/2(2m_1+1)-r)}\|\varphi_k\|_t, \]
\[ r=0,1,\ldots,n_0=\left[n-\frac{n}{2(2m_1+1)}\right]. \]
\[ \text{V.}\qquad |k|^r|u_k(t)|\le C_5 |k|^{-(n-n/2(2m_1+1)-r)}\|\varphi_k\|_t, \]
\[ r=0,1,\ldots,n_0=\left[n-\frac{n}{2(2m_1+1)}\right]. \]
Remark. In essence, in each of the cases one should write a pair of inequalities. For example, in case I, in addition to the displayed inequality, an inequality of the form
\[ |u_k^{(r)}(t)|\le C_1' \alpha_k^{-(2m_1-1/2-r)}\|\varphi_k\|_t,\qquad \alpha_k=|k|^{n/m},\quad r=0,1,\ldots,2m_1-1 \]
is valid.
In the estimates given, \(C\) does not depend on \(k\), the square brackets denote the integer part, and
\[ \|\varphi_k\|_t^2=\int_0^T |\varphi_k(t)|^2\,dt. \]
For \(k=0\) the validity of the estimates of Lemma 1 is obvious (in some cases, it may be necessary to impose additional conditions in order to exclude polynomials in \(t\) from consideration). For \(k\ne0\) the estimates of Lemma 1 are obtained from the formula for the solution of problem (4), (5):
\[ \begin{aligned} u_k(t)=&-\frac{(-1)^m}{\alpha_k^{m-1} W_m(\omega)} \sum_{s=1}^{m} e^{\alpha_k \omega_s t} \Biggl[ (-1)^s W_{m-1}(\omega\ne\omega_s) \int_{0}^{t} e^{-\alpha_k\omega_s\tau}\varphi_k\,d\tau \\ &\quad -\frac{1}{\Delta(\alpha_kT)} \sum_{\substack{p'=1\\ p'\ne s}}^{m} (-1)^{\mathscr p_{p'}'}D_{m-1}(\omega\ne\omega_{p'}) \\ &\quad\times \sum_{q=1}^{m}(-1)^q W_{m-1}(\omega\ne\omega_q) D_\nu(\omega_q,\omega_{p'}) e^{\alpha_kT(\omega_q+|\omega_{p'}|)} \int_{0}^{T} e^{-\alpha_k\omega_q\tau}\varphi_k\,d\tau \Biggr], \end{aligned} \tag{6} \]
where \(\alpha_k=|k|^{n/m}\); \(\alpha_k\omega_1,\ldots,\alpha_k\omega_m\) are the roots of the characteristic equation for (4); \(W_m(\omega)\) is the Vandermonde determinant of order \(m\) formed from the numbers \(\omega=(\omega_1,\ldots,\omega_m)\), and \(W_{m-1}(\omega\ne\omega_s)\) from the numbers \(\omega_1,\ldots,\omega_{s-1},\omega_{s+1},\ldots,\omega_m\); \(p=(p',p_\nu)\); \(p'=(p_1,\ldots,p_{\nu-1})\); \(\omega_p=(\omega_{p'},\omega_{p_\nu})\); \(\omega_{p'}=(\omega_{p_1},\ldots,\omega_{p_{\nu-1}})\); \(|p'|=p_1+\cdots+p_{\nu-1}\); \(|p|=|p'|+p_\nu\); \(|\omega_{p'}|=\omega_{p_1}+\cdots+\omega_{p_{\nu-1}}\); \(\omega_p=|\omega_{p'}|+\omega_{p_\nu}\); \(p_i\), \(p_j\), and \(q\) are pairwise distinct;
\[ \mathscr p_p=m\nu-\frac12\nu(\nu-1)+|p|; \]
\[ \mathscr p_{p'}'=m\nu-\frac12\nu(\nu-1)+|p'|+s; \qquad \Delta(\alpha_kT)= \sum_{p=1}^{m} (-1)^{\mathscr p_p} D_\nu(\omega_p) D_{m-\nu}(\omega\ne\omega_p) e^{\alpha_kT|\omega_p|}, \]
where \(D_\nu\) and \(D_{m-\nu}\) are determinants of order \(\nu\) and \(m-\nu\), respectively, for example
\[ D_\nu(\omega_p)= \left| \begin{array}{ccc} \omega_{p_1}^{r_1} & \cdots & \omega_{p_\nu}^{r_1}\\ \cdots & \cdots & \cdots\\ \omega_{p_1}^{r_\nu} & \cdots & \omega_{p_\nu}^{r_\nu} \end{array} \right|. \]
Following (4) (p. 78), one can show that \(\Delta(\alpha_kT)\ne0\).
Putting, as usual,
\[ (u,v)=\int_V u\bar v\,dV; \qquad |u,H|^2=(u,u) \]
and completing \(P_{t,x}^{m+1,n+1}\) with respect to the introduced norm, we obtain a Hilbert space \(H\) of square-summable functions.
It follows from Lemma 1 that on the set \(u\in P_{t,x}^{m+1,n+1}\) satisfying (3), \((\Gamma)\), one can introduce the scalar product
\[ \langle u,v\rangle=(Lu,Lv). \tag{7} \]
The Hilbert space obtained by completing \(P_{t,x}^{m+1,n+1}\) in the metric (8) will be denoted by \(\mathscr H\). The inequalities of Lemma 1 show that the embedding
\[ \mathscr H\subseteq H_{t,x}^{m-1,n_0}, \]
is valid, where \(H_{t,x}^{m-1,n_0}\) is the Hilbert space of functions having generalized partial derivatives with respect to \(t\) of order \((m-1)\) and with respect to \(x\) of order \(n_0\).
A function \(u\in\mathscr H\) will be called a strong solution of problem (1), \((\Gamma)\), if there exists a sequence \(\{u_i\}\) of functions from \(P_{t,x}^{m+1,n+1}\), satisfying the conditions \((\Gamma)\), for which
\[ |u_i-u,\mathscr H|\to0, \qquad |Lu_i+L_1(t,x)u_i-f,H|\to0 \quad\text{as } i\to\infty. \]
Lemma 2. For every function \(\varphi\in H\) there exists a unique strong solution of problem (3), \((\Gamma)\).
The validity of this lemma follows from Lemma 1 and from the possibility of approximating \(\varphi\in H\) by finite sums of the form \(\sum \varphi_k e^{ik\pi x/l}\) with smooth functions \(\varphi_k\).
Theorem 1. Problem (1), (Γ) is regularly solvable, i.e., the three Fredholm theorems are valid for it.
Remark. After the substitution
\[
u(t,x)=v(t,x)\exp\left[-\frac1m\int_0^t a_{m-1}(\tau,x)\,d\tau\right]
\]
for the function \(v(t,x)\) one obtains an equation of the same form as (1), but with \(a_{m-1}(t,x)=0\). Therefore it is sufficient to carry out the proof of Theorem 1 for the equation
\[
Lu+L_2(t,x)u=f,
\tag{8}
\]
where for \(L_2(t,x)\)
\[
r=\frac{r_1}{m-1}+\frac{r_2}{n_0}<1.
\]
From Lemma 2 there follows the existence of the operator \(L^{-1}:H\to\mathcal H\), and hence also the existence of the operator \(A=L^{-1}L_2:\mathcal H\to\mathcal H\). In this case problem (9), (Γ) will be equivalent to the operator equation in the space \(\mathcal H\)
\[
u+Au=g\equiv L^{-1}f.
\tag{9}
\]
Using the inequalities of Lemma 1, the embedding theorems \((^6)\) for \(m>1,\ n>1\), and proceeding as in \((^5)\) for \(m=1\) or \(n=1\), one can show that \(A\) is a completely continuous operator, i.e., equation (9) is of Fredholm type, which proves Theorem 1.
Theorem 2. For sufficiently small \(T\), for every function \(f\in H\) there exists a unique strong solution of (1), (Γ).
We indicate in conclusion a method of constructing examples showing that in each case going outside the described class of problems leads to ill-posedness or instability.
Example. Let
\[
Lu\equiv \frac{\partial^4u}{\partial t^4}+\frac{\partial^4u}{\partial x^4}
=e^{ik\pi x/l}\equiv f^{(k)}.
\]
For this equation, which is elliptic, the Dirichlet problem is known to be well posed. Let us now impose on the function \(u(t,x)\) the conditions
\[
u(T,x)=u(0,x)=u_t(0,x)=u_{t^2}(0,x)=0
\]
and the periodicity condition in \(x\).
The solution of this problem has the form
\[
u_k(t)=\frac{l^4}{\pi^4 k^4}e^{ik\pi x/l}
\left[
1-\operatorname{ch}\alpha_k t\cos\alpha_k t
-\frac{1-\operatorname{ch}\alpha_k T\cos\alpha_k T}{\Delta(\alpha_kT)}
\Delta(\alpha_k t)
\right],
\]
where
\[
\alpha_k=\frac{\pi k}{l\sqrt2},\qquad
\Delta(\alpha_k t)=\operatorname{ch}\alpha_k t\sin\alpha_k t-\operatorname{sh}\alpha_k t\cos\alpha_k t,\qquad
\Delta(\alpha_kT)=\Delta(\alpha_k t)\big|_{t=T}.
\]
For \(k\to\infty\)
\[
k^4\Delta(\alpha_kT)=O\left[k^4e^{\alpha_kT}\cos(\alpha_kT-3\pi/4)\right].
\]
From the formula for the solution we obtain that, for \(0<t=t_0<T,\ k\to\infty\), and
\[
\alpha_kT-3\pi/4\ne \pi/2+n\pi,\qquad n=0,1,2,\ldots,
\]
\(\lvert u^{(k)}\rvert\to\infty\), whereas \(\lvert f^{(k)}\rvert=1\). On the other hand, by changing \(T\) and \(l\) one can obtain
\[
\alpha_kT-3\pi/4=\pi/2+n\pi,
\]
and then, as is not hard to show, the homogeneous problem has a nontrivial solution, i.e., uniqueness is violated. Thus the chosen problem possesses ill-posedness similar to the Cauchy problem for the Laplace operator and at the same time instability analogous to the instability of the Dirichlet problem for the wave equation.
In conclusion the author takes the opportunity to express gratitude to A. A. Dezin for posing the problem and for his constant attention.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
7 V 1964
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