Full Text
Doklady Akademii Nauk SSSR
1958, Volume 122, No. 4
MATHEMATICS
Academician S. L. SOBOLEV
ON MIXED PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH TWO INDEPENDENT VARIABLES
Consider an equation with constant coefficients
\[ \frac{\partial^n u}{\partial t^n} + \sum_{k<n,\; l\le m} A_{k,l}\, \frac{\partial^{k+l}u}{\partial t^k \partial x^l} =f \tag{1} \]
with two independent variables and one unknown function, satisfying Petrovskii’s condition of uniform correctness of the Cauchy problem. The structure of this equation was analyzed by us in the preceding note \((^1)\). We shall study this equation in the domains:
\[ \begin{aligned} \text{a)}\;& -\infty < x < +\infty,\quad 0\le t < +\infty;\\ \text{b)}\;& 0\le x < +\infty,\quad 0\le t < +\infty;\\ \text{c)}\;& 0\le x \le 1,\quad 0\le t < +\infty. \end{aligned} \tag{2} \]
under the initial conditions
\[ u\big|_{t=0} = \frac{\partial u}{\partial t}\bigg|_{t=0} = \cdots = \frac{\partial^{n-1}u}{\partial t^{n-1}}\bigg|_{t=0} =0 \tag{3} \]
and under certain homogeneous conditions on the boundaries:
\[ \sum_{i=1}^{m-1} g_i^{(s)} \frac{\partial^i u}{\partial x^i}\bigg|_{x=0} =0, \quad s=1,2,\ldots,q_-, \tag{4} \]
for b) and c), and
\[ \sum_{i=1}^{m-1} h_i^{(s)} \frac{\partial^i u}{\partial x^i}\bigg|_{x=1} =0, \quad s=1,2,\ldots,q_+, \tag{5} \]
for c).
We shall deal with the conditions for solvability of such a problem and with the correctness of its formulation. Let us note that certain analogous formulations occurred in the work of M. I. Vishik and L. A. Lyusternik \((^2)\) as one of the links in their theory. We shall consider this question in general form, in its own right.
We shall seek the solution of the problems posed by means of the Laplace transform. Let
\[ \tilde u(x,\lambda) = \int_0^\infty e^{-\lambda t}u(x,t)\,dt, \qquad \tilde f(x,\lambda) = \int_0^\infty e^{-\lambda t}f(x,t)\,dt. \tag{6} \]
Formula (6) makes sense, for example, if \(|\tilde u|<e^{Mt}\) and \(|\tilde f|<e^{Mt}\), where \(M\) is some constant. The function \(\tilde u(x,\lambda)\) will be a solution of the ordinary equation
\[ L_\lambda \tilde u \equiv \lambda^n + \sum_{k<n,\; l\le m} A_{k,l}\lambda^k \frac{d^l\tilde u}{dx^l} = \tilde f, \tag{7} \]
which may also be written as
\[ a_0(\lambda)\frac{d^m\tilde u}{dx^m} +a_1(\lambda)\frac{d^{m-1}\tilde u}{dx^{m-1}} +\cdots+a_m(\lambda)\tilde u=\tilde f . \tag{8} \]
The solution of (7), with the aid of Green’s function, is expressed in the form
\[ \tilde u(x,\lambda)=\int_{-\infty}^{+\infty} G(x,x_1,\lambda)\tilde f(x_1,\lambda)\,dx_1 . \tag{9} \]
Let us write the differential equation for the Green’s function. For case a) this equation will be
\[ L_\lambda G=\delta(x-x_1), \tag{10} \]
where \(\delta(x-x_1)\) is the generalized Dirac function. Let
\[
\left.\frac{d^rG^{(б)}}{dx^r}\right|_{x=0}=A_r(\lambda).
\]
Extend the function \(G^{(б)}\) by zero for negative \(x\). Computing \(L_\lambda G^{(б)}\) by the usual method, we obtain the formula
\[ L_\lambda G^{(б)} =\sum_{k=0}^{m-1} C_k\delta^{(k)}(x)+\delta(x-x_1), \qquad \text{where }\quad C_k=\sum_{j=0}^{m-k-1} a_jA_{m-k-j-1}. \tag{11} \]
In exactly the same way, for the function \(G^{(в)}\), extended by zero outside the interval \(0\leq x\leq 1\), we obtain:
\[ L_\lambda G^{(в)} =\sum_{k=0}^{m-1} C_k\delta^{(k)}(x) +\sum_{k=0}^{m-1} D_k\delta^{(k)}(x-1) +\delta(x-x_1), \tag{12} \]
where \(C_k\) is expressed by formula (11), and \(D_k\) by the formula
\[ D_k=-\sum_{j=0}^{m-k-1} a_jB_{m-k-j-1}, \qquad \text{where }\quad \left.\frac{d^rG^{(в)}}{dx^r}\right|_{x=1}=B_r(\lambda). \tag{13} \]
Let us apply to the determination of \(G\), in all cases, one more Laplace transformation with respect to the variable \(x\). We have
\[ \widehat{L_\lambda G}=\Delta(\lambda,\alpha)\,\tilde G, \qquad \text{where }\quad \tilde G=\int_{-\infty}^{+\infty} e^{-\alpha x}G(x)\,dx . \tag{14} \]
Here by \(\Delta(\lambda,\alpha)\) is denoted the polynomial
\[
\Delta(\lambda,\alpha)=\lambda^n+\sum_{\substack{k<n\\ l<m}} A_{k,l}\lambda^k\alpha^l .
\]
The expression \(L_\lambda G\) is easily computed by means of formulas (10), (11), or (12); we shall have:
\[ \widehat{L_\lambda G^{(a)}}=e^{-\alpha x_1},\qquad \widehat{L_\lambda G^{(б)}}=e^{-\alpha x_1}+\sum_{k=0}^{m-1} C_k\alpha^k, \]
\[ \widehat{L_\lambda G^{(в)}}=e^{-\alpha x_1} +\sum_{k=0}^{m-1} C_k\alpha^k +e^{-\alpha}\sum_{k=0}^{m-1} D_k\alpha^k . \tag{15} \]
Consequently:
\[ \widehat{G^{(a)}}=\frac{e^{-\alpha x_1}}{\Delta(\lambda,\alpha)},\qquad \widehat{G^{(б)}}= \frac{e^{-\alpha x_1}+\sum_{k=0}^{m-1} C_k\alpha^k} {\Delta(\lambda,\alpha)}, \]
\[ \widehat{G^{(в)}}= \frac{e^{-\alpha x_1}+\sum_{k=0}^{m-1} C_k\alpha^k +e^{-\alpha}\sum_{k=0}^{m-1} D_k\alpha^k} {\Delta(\lambda,\alpha)} . \tag{16} \]
The first formula (16) must express the Laplace transform of \(G^{(a)}\) and, consequently, must represent a regular function in some strip \(M_1<\xi<M_2\), where \(\alpha=\xi+i\eta\). Choosing different strips that do not contain roots of the denominator, we obtain different values for \(\widetilde{G}^{(a)}\). The Mellin integral makes it possible to reconstruct \(\widetilde{G}^{(a)}\) from its image, reducing it to residues at the roots of \(\Delta(\lambda,\alpha)\). The investigation shows that only one of the values obtained, namely the one corresponding to the strip containing the imaginary axis, gives the desired solution of the problem, decreasing for large values of \(\lambda\). Thus the Cauchy problem is solved.
The function \(\widetilde{G}^{(b)}(\alpha)\) must be the image of a function \(G^{(b)}(x)\) equal to zero for \(x<0\). Computing this latter function by means of the Mellin formula and again taking into account the requirement of decrease for large values of \(\lambda\), we are convinced that this is possible only in the case when \(\widetilde{G}^{(b)}(\alpha)\) is regular at all roots \(\gamma_s\) of the equation \(\Delta(\lambda,\alpha)\) lying to the right of the imaginary axis \(\alpha\).
We decompose \(\Delta(\lambda,\alpha)\) into two factors
\[ \Delta(\lambda,\alpha)=\Delta_1(\lambda,\alpha)\Delta_2(\lambda,\alpha),\quad \Delta_1(\lambda,\alpha)=\prod_{s=1}^{r_-}(\alpha-\beta_s),\quad \Delta_2(\lambda,\alpha)=\prod_{s=1}^{r_+}(\alpha-\gamma_s), \tag{17} \]
\(\beta_s(\lambda)\) are the roots of \(\Delta(\lambda,\alpha)\) situated in the left half-plane, and \(\gamma_s(\lambda)\) are the roots of this polynomial situated in the right half-plane \(\alpha\). We shall call the number of roots \(\beta_s(\lambda)\) the number of left influences, and the number of roots \(\gamma_s(\lambda)\) the number of right influences for equation (1). Let us construct the remainder upon division of \(\widetilde{G}^{(b)}\) by \(\Delta_2(\lambda,\alpha)\). For this it is enough to consider a contour \(C_\gamma\) enclosing all roots \(\gamma_s\) and containing no \(\beta_s\), and to compute the function
\[ R_\gamma=\Delta_2(\lambda,\alpha)\left(\widetilde{G}^{(b)}-\frac{1}{2\pi i}\int_{C_\gamma}\frac{\widetilde{G}^{(b)}(\alpha')\,d\alpha'}{\alpha'-\alpha}\right). \tag{18} \]
This function will be a polynomial of degree \(r_+\) in \(\alpha\), which serves as the required remainder. The coefficients \(R_\gamma\) are linear functions of \(C_0,C_1,\ldots,C_{m-1}\). Equating them to zero, we obtain a system of linear equations for \(C_j\). These equations are independent, since each of the first coefficients \(C_0,C_1,\ldots,C_{r_+}\) enters into one and only one of these equations with coefficient equal to unity. In order that our problem have a solution, one must have still \(r_-\) linear relations independent of the preceding ones. Such relations are obtained from (4) in the form
\[ \sum_{i=1}^{m-1} g_i^{(s)}A_i=0,\qquad s=1,2,\ldots,q_-. \tag{19} \]
The quantities \(C_j\) found from the resulting system will decrease exponentially for large \(\lambda\). Examples show that it is possible that equations (19), in different cases, will turn out to be either dependent or independent of the rest. We obtain the main theorem.
Theorem. The mixed problem in domain b), generally speaking, is solvable and has a unique regular solution if the number of conditions \(q_-\) is equal to the number of left influences \(r_-\). It is solvable, in particular, if the conditions have the form:
\[ \left.\frac{\partial^k u}{\partial x^k}\right|_{x=0}=0,\qquad k=0,1,\ldots,r_-. \tag{20} \]
Another particular case of solvability is the case when \(Lu\) is an elementary Petrovskii operator.
We pass to case c). It is easy to see that the function \(\widetilde{G}^{(\mathrm{c})}(\alpha)\) can serve as a prototype for \(G^{(\mathrm{c})}(x)\), equal to zero outside \((0,1)\), only in the case when it is regular in the whole \(\alpha\)-plane. The formula
\[ R=\Delta(\lambda,\alpha)\left(\widetilde{G}^{(\mathrm{c})}(\alpha)-\frac{1}{2\pi i}\int_C \frac{G^{(\mathrm{c})}(\alpha')}{\alpha'-\alpha}\,d\alpha'\right) \tag{21} \]
gives the remainder upon division of \(\widetilde{G}^{(\mathrm{c})}(\alpha)\) by \(\Delta(\lambda,\alpha)\), which is a polynomial of degree \(m-1\). All coefficients of this polynomial must be equal to zero; hence we obtain \(m\) relations for determining the constants \(C_k\) and \(D_k\). Equations (4) and (5) give another \(m\) relations. If these relations are independent, then all the required constants can be found. It remains to consider the conditions under which the quantities \(C_k\) and \(D_k\) thus found will decrease in the right half-plane for large values of \(\lambda\).
Let \(C\) be the vector with components \(C_0,C_1,\ldots,C_{m-1}\), and \(D\) the vector with components \(D_0,D_1,\ldots,D_{m-1}\). The equations obtained by setting \(R\) equal to zero can be represented in another form, by reducing them to equations expressing the vanishing at infinity of \(\widetilde{G}^{(\mathrm{c})}\) at each of the roots of \(\Delta(\lambda,\alpha)\) separately. Writing separately the terms corresponding to \(\beta_s\) and \(\gamma_s\), we obtain the system
\[ \sum_{i=1}^{m} c_i^{(s)} C_i+\sum_{i=1}^{m} d_i^{(s)} e^{-\beta_s} D_i =F_s e^{-\beta_s x_3},\qquad s=1,2,\ldots,r_+; \tag{22} \]
\[ \sum c_i^{(s)} C_i+\sum_{i=1}^{m} d_i^{(s)} e^{-\gamma_s} D_i =F_s e^{-\gamma_s x_1},\qquad s=r_+ +1, r_+ +2,\ldots,m. \tag{23} \]
Multiplying each of equations (23) by \(e^{-\gamma_s x_1}\) and adjoining to (22) and (23) the equations obtained from (4) and (5), we put this system in the form
\[ X_1 C+Y_1 D=F_1,\qquad Y_2 C+X_2 D=F_2, \tag{24} \]
where the matrix \(X_1\) consists of \(m\) columns and \(r_+ + q_-\) rows, the matrix \(X_2\) of \(m\) columns and \(r_- + q_+\) rows; moreover, the elements of \(X_1\) and \(X_2\), as \(\lambda\) increases, remain finite. The matrices \(Y_1\) and \(Y_2\), as well as the matrices \(F_1\) and \(F_2\), decrease exponentially for large \(\lambda\) such that \(\sigma>\sigma_0\). It is easy to verify that, if \(q_-=r_-\), \(q_+=r_+\), and \(X_1^{-1}, X_2^{-1}\) do not tend to zero, then system (24) has a solution that converges for large \(\lambda\) and satisfies all the imposed conditions, representable in the form
\[ \begin{aligned} C&=X_1^{-1}F_1-X_1^{-1}Y_1X_2^{-1}F_2+X_1^{-1}Y_1X_2^{-1}Y_2X_1^{-1}F_1-\cdots,\\ D&=X_2^{-1}F_2-X_2^{-1}Y_2X_1^{-1}F_1+X_2^{-1}Y_2X_1^{-1}Y_1X_2^{-1}F_2+\cdots . \end{aligned} \]
If \(q_-\ne r_-\) and, consequently, \(q_+\ne r_+\), then the solution of system (24) gives, generally speaking, \(C\) and \(D\) growing exponentially together with \(\lambda\). We obtain the theorem:
Theorem. Problem c), generally speaking, is solvable if the number of conditions at the left end \(q_-\) is equal to the number \(r_-\) of left influences, and the number of conditions \(q_+\) at the right end is equal to the number \(r_+\) of right influences.
In particular, as in the case of problem b), the problem is always solvable for the case of the simplest conditions of the form (20), and also for elementary Petrovskii operators.
Received
17 VII 1958
References
- S. L. Sobolev, Dokl. Akad. Nauk SSSR 121, No. 4 (1958).
- M. I. Vishik, L. A. Lyusternik, Uspekhi Mat. Nauk 12, issue 5 (77), 3 (1957).