Abstract
Full Text
V. A. SOLONNIKOV
ON ESTIMATES OF FUNDAMENTAL MATRICES FOR GENERAL PARABOLIC SYSTEMS WITH CONSTANT COEFFICIENTS
(Presented by Academician V. I. Smirnov on 13 IV 1964)
The present work is devoted to an estimate of the fundamental matrix by means of which the solution of a general boundary-value problem for a homogeneous parabolic system with constant coefficients in a half-space is constructed.
Consider in the half-space \(E_{n+1}^{+}\), whose points have coordinates \(x_1,\ldots,x_n,t\), with \(x_n \geqslant 0\), the problem
\[ \mathcal{L}_0\left(\frac{\partial}{\partial x},\frac{\partial}{\partial t}\right)u(x,t)=0,\qquad B_0\left(\frac{\partial}{\partial x},\frac{\partial}{\partial t}\right)u(x,t)\bigg|_{x_n=0}=\Phi(x',t), \tag{1} \]
where \(x=(x_1,\ldots,x_n)\), \(x'=(x_1,\ldots,x_{n-1})\), \(\mathcal{L}_0\) and \(B_0\) are matrices whose elements are linear differential operators \(l^0_{kj}(\partial/\partial x,\partial/\partial t)\) and \(B^0_{qj}(\partial/\partial x,\partial/\partial t)\) with complex coefficients. We assume that the system under consideration satisfies the parabolicity condition formulated in (1), and, moreover, contains only highest-order terms, i.e.: 1) there exist integers \(s_k\) and \(t_j\) \((k,j=1,\ldots,m,\ \max_k s_k=0)\) and an integer \(b>0\) such that
\[ l^0_{kj}(i\xi\lambda,p\lambda^{2b})=\lambda^{s_k+t_j}l^0_{kj}(i\xi,p), \]
where \(i=\sqrt{-1}\) and \(\xi=(\xi_1,\ldots,\xi_n)\); 2) the roots \(p_s\) of the polynomial \(L(i\xi,p)=\det \mathcal{L}_0(i\xi,p)\) satisfy the condition
\[ \operatorname{Re} p_s \leqslant -\delta \xi^{2b} \qquad (\delta>0,\ \xi^2=\xi_1^2+\cdots+\xi_n^2). \]
From condition 1) it follows that the polynomial \(L(i\xi,p)\) is homogeneous:
\[ L(i\xi\lambda,p\lambda^{2b}) = \lambda^{\sum_{k=1}^{m}(s_k+t_k)}L(i\xi,p), \]
and from condition 2), that \(L(0,p)=\gamma p^r,\ \gamma\neq 0\), and
\[ \sum_{k=1}^{m}(s_k+t_k)=2br=2R,\qquad R>0. \]
The polynomial \(L(i\zeta,i\tau,p)\), where \(\zeta=(\zeta_1,\ldots,\zeta_{n-1})\), has exactly \(R\) roots in \(\tau\) with positive imaginary part (we denote them by \(\tau_s^+\)) and \(R\) with negative imaginary part, if \(\zeta_j\) are real and
\[ \operatorname{Re} p \geqslant -\delta_1\zeta^{2b},\qquad |p|^2+\zeta^{4b}>0 \qquad (\delta_1<\delta). \tag{2} \]
It is also assumed that there exist integers \(\sigma_q\) \((q=1,\ldots,R)\) such that
\[ B^0_{qj}(i\xi\lambda,p\lambda^{2b})=\lambda^{\sigma_q+t_j}B^0_{qj}(i\xi,p) \]
and the following algebraic complementing condition is fulfilled: the rows of the matrix
\[ A(i\zeta,i\tau,p)=B_0(i\zeta,i\tau,p)\,\mathcal{L}_0^{\prime}(i\zeta,i\tau,p) \quad (\mathcal{L}_0^{\prime}=L\mathcal{L}_0^{-1}) \]
as polynomials in \(\tau\) are linearly independent modulo the polynomial
\[ M^{+}=\prod_{s=1}^{k}\bigl(\tau-\tau_s^{+}(\zeta,p)\bigr) \]
for any real \(\zeta\) and \(p\) satisfying condition (2).
Applying in (1) the Fourier transform with respect to the variables \(x_1,\ldots,x_{n+1}\) and the Laplace transform with respect to \(t\), it is easy to show that the solution
\(u=(u_1,\ldots,u_m)\) of problem (1) is expressed in terms of the given vector \(\Phi=(\Phi_1,\ldots,\Phi_R)\) by the formula
\[ u_j(x,t)=\sum_{q=1}^{R}\int_{-\infty}^{+\infty}d\tau \int_{E_{n-1}} G_{jq}(x'-y',x_n,t-\tau)\Phi_q(y',\tau)\,dy', \]
where
\[ G_{jq}(x,t)\frac{1}{(2\pi)^{n-1}2\pi i} \int_{E_{n-1}} e^{i(x',\zeta)}d\zeta \int_{\sigma-i\infty}^{\sigma+i\infty} e^{p t}\widetilde{G}_{jq}(\zeta,p,x_n)\,dp, \tag{3} \]
where \(\sigma>-\delta_1\zeta^{2b}\), and \(\widetilde{G}_{jq}\) is the solution of the following boundary-value problem for a system of ordinary differential equations:
\[ \sum_{j=1}^{m} l_{kj}^{0}\left(i\zeta,\frac{d}{dx_n},p\right)\widetilde{G}_{jq}(\zeta,p,x_n)=0 \quad (k=1,\ldots,m); \tag{4} \]
\[ \sum_{j=1}^{m} B_{sj}^{0}\left(i\zeta,\frac{d}{dx_n},p\right) \widetilde{G}_{jq}(\zeta,p,x_n)\bigg|_{x_n=0} =\delta_{sq},\qquad \widetilde{G}_{jq}\xrightarrow[x_n\to+\infty]{}0. \tag{5} \]
Here \(s,q=1,\ldots,R\); \(\delta_{sq}\) is the Kronecker symbol.
The question of estimating the functions (3) for systems parabolic in the sense of Petrovskii was studied in works of T. Ya. Zagorskii and S. D. Eidelman. T. Ya. Zagorskii \((^2)\) proposed for this purpose a number of techniques connected with deformation of the contours along which the integration in (3) is performed. However, in \((^2)\) these estimates were carried out without sufficient justification, since one cannot consider it established in \((^2)\) that \(\widetilde{G}_{jq}\) is analytic in the variables \(\zeta_j\) and \(p\) under general assumptions. In the notes of S. D. Eidelman \((^3,^4)\) more precise estimates of the functions \(\widetilde{G}_{jq}\) were formulated than in \((^2)\) (they are extremely sharp). In \((^4)\) it is indicated that the analyticity of \(\widetilde{G}_{jq}\) can be derived from results of V. P. Mikhailov \((^5)\) on the solvability of the first boundary-value problem for Petrovskii-parabolic systems and from an algebraic theorem of Ya. B. Lopatinskii \((^6)\). In obtaining estimates of solutions of boundary-value problems for general parabolic systems \((^1)\), the analyticity property of the functions \(\widetilde{G}_{jq}\) was also used. It was established with the help of a special additional assumption that the system determined by the transposed matrix \(\mathscr{L}_0^*\) has at least one boundary-value problem satisfying the same conditions as problem (1). We shall establish the analyticity of the functions \(\widetilde{G}_{jq}\) without this assumption.
First of all, we note that problem (4)—(5), when the complementing condition is satisfied, has only one solution. This assertion was proved in \((^7)\). Let now \(A'\) be the matrix with elements
\[ A'_{qj}(\zeta,p,\tau)=\sum_{s=1}^{R}\alpha_s^{(q,j)}(\zeta,p)\tau^{s-1}, \]
which are the remainders obtained by dividing the elements of the matrix \(A\) by \(M^+\). The complementing condition is equivalent to the following assertion: if \(\operatorname{Im}\zeta=0\) and \(p\) satisfies conditions (2), then the rank of the matrix
\[ \mathfrak{A}= \begin{pmatrix} \alpha_1^{(1,1)} \ldots \alpha_R^{(1,1)} \alpha_1^{(1,2)} \ldots \alpha_R^{(1,2)} \ldots \alpha_1^{(1,m)} \ldots \alpha_R^{(1,m)} \\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot \\ \alpha_1^{(R,1)} \ldots \alpha_R^{(R,1)} \alpha_1^{(R,2)} \ldots \alpha_R^{(R,2)} \ldots \alpha_1^{(R,m)} \ldots \alpha_R^{(R,m)} \end{pmatrix} \]
is equal to \(R\).
Arguing as in \((^7,{}^8)\), it is easy to show that
\[ \widetilde{G}_{jq}(\xi,p,x_n) = \frac{1}{2\pi i} \sum_{k=1}^{m} \int_{\gamma^+} \frac{L_{kj}(i\xi,i\tau,p)\,N_{kq}(\xi,p,\tau)} {M^+(\xi,p,\tau)} e^{i\tau x_n}\,d\tau . \tag{6} \]
Here \(\gamma^+\) is a contour enclosing all roots \(\tau_s^+\); \(L_{kj}\) is the algebraic complement of the element \(l_{kj}^{0}\) in \(\mathcal{L}_0\), and
\[ N_{kq}(\xi,p,\tau) = \sum_{s=1}^{R} \beta_s^{(k,q)}(\xi,p)\,M_{R-s}^{+}(\xi,p,\tau), \]
where \(\beta_s^{(k,q)}\) are the elements of the right inverse matrix \(\mathfrak{B}\) to the matrix \(\mathfrak{A}\):
\[ \mathfrak{B}^{*} = \left( \begin{array}{ccccccccccc} \beta_{1}^{(1,1)} & \ldots & \beta_{R}^{(1,1)} & \beta_{1}^{(2,1)} & \ldots & \beta_{R}^{(2,1)} & \ldots & \beta_{1}^{(m,1)} & \ldots & \beta_{R}^{(m,1)} \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \beta_{1}^{(1,R)} & \ldots & \beta_{R}^{(1,R)} & \beta_{1}^{(2,R)} & \ldots & \beta_{R}^{(2,R)} & \ldots & \beta_{1}^{(m,R)} & \ldots & \beta_{R}^{(m,R)} \end{array} \right)^{*}, \]
and \(M_q^+\) are polynomials related to \(M^+\) in the following way: if
\[ M^+ = \sum_{s=1}^{R} m_s(\xi,p)\,\tau^{R-s}, \quad\text{then}\quad M_q^+ = \sum_{s=1}^{q} m_s(\xi,p)\,\tau^{q-s} \quad (q=1,\ldots,R). \]
The choice of the matrix \(\mathfrak{B}\) may be nonunique, but since \(\widetilde{G}_{jq}\) is unique, this nonuniqueness plays no role. We shall establish all the necessary properties of the functions \(\widetilde{G}_{qj}\) by choosing as \(\mathfrak{B}\) various right inverse matrices.
Let \(\Delta_1,\ldots,\Delta_N(\xi,p)\) be all non-identically-zero determinants of the square matrices \(\mathfrak{A}_i(\xi,p)\) of order \(R\), composed of columns of the matrix \(\mathfrak{A}\). By the complementarity condition, for the values of \(\xi\) and \(p\) under consideration they have no common zeros. They are homogeneous:
\[ \Delta_i(\lambda\xi,\lambda^{2b}p) = \lambda^{k_i}\Delta_i(\xi,p) \quad (\lambda>0). \]
Choose numbers \(\chi_i\ge 1\) so that \(\Delta_i^{\chi_i}\) have the same degree of homogeneity. Let
\[ D(\xi,p) = \sum_{i=1}^{N} \Delta_i^{\chi_i}\,\overline{\Delta}_i^{\chi_i} = \sum_{i=1}^{N} |\Delta_i|^{2\chi_i}, \qquad \Delta^{(i)}(\xi,p) = \frac{\Delta_i^{\chi_i-1}\overline{\Delta}_i^{\chi_i}}{D}. \]
We have \(\sum_{i=1}^{N}\Delta_i\Delta^{(i)}=1\). Now one may put
\[ \mathfrak{B} = \sum_{i=1}^{N} \Delta^{(i)}\mathfrak{B}_i, \tag{7} \]
where \(\mathfrak{B}_i\) is the matrix composed of zero rows and rows of the matrix \(\widehat{\mathfrak{A}}_i\), reciprocal to \(\mathfrak{A}_i\), in such a way that \(\mathfrak{A}\mathfrak{B}_i=\Delta_i E\) (\(E\) is the identity matrix of order \(R\)).
If, as \(\beta_s^{(k,q)}\), one takes the elements of the matrix (7), then from (6) it is easy to see that \(\widetilde{G}_{jq}\) are smooth single-valued functions of the variables \(\xi_i\) and \(p\), analytic in \(\xi_j\) in a neighborhood of the real axis for every \(p\) satisfying condition (2). Moreover, \(\widetilde{G}_{jq}\) satisfy the necessary homogeneity conditions
\[ \widetilde{G}_{jq}\left(\lambda\xi,\lambda^{2b}p,\frac{x_n}{\lambda}\right) = \lambda^{-\sigma_q-t_j}\widetilde{G}_{jq}(\xi,p,x_n). \]
\[ \text{* For economy of space we give the formula for the transposed matrix } \mathfrak{B}^{*}. \]
It remains to show that \(\widetilde G_{iq}\) are analytic in \(p\) in the domain (2) under consideration for every real \(\zeta\). To this end fix an arbitrary \(p_0\) in this domain. Let
\[ D'(\zeta,p,p_0)=\sum_{i=1}^{N}\Delta_i^{\varkappa_i}(\zeta,p)\,\overline{\Delta_i}^{\,\varkappa_i}(\zeta,p_0), \]
\[ \Delta'^{(i)}(\zeta,p,p_0)= \frac{\Delta_i^{\varkappa_i-1}(\zeta,p)\,\overline{\Delta_i}^{\,\varkappa_i}(\zeta,p_0)} {D'(\zeta,p,p_0)}, \]
\[ \mathfrak B'=\sum_{i=1}^{N}\Delta'^{(i)}(\zeta,p,p_0)\,\mathfrak B_i(\zeta,p). \]
Since \(D'(\zeta,p_0,p_0)=D(\zeta,p_0)>0\), we have \(D'(\zeta,p,p_0)\ne0\) in a neighborhood of the point \(p_0\), and for \(p\) close to \(p_0\) the elements of the matrix \(\mathfrak B'\) may be taken as \(\beta_{\zeta}^{(k,q)}\). But now, since \(\overline{\Delta_i}\) do not depend on \(p\), it is clear that \(\widetilde G_{iq}\) are analytic in \(p\) in a neighborhood of \(p_0\). Since \(p_0\) is arbitrary, it follows that \(\widetilde G_{iq}\) is analytic in \(p\) throughout the whole domain (2).
Besides \(G_{iq}\), it is useful to consider the functions
\[ G_{iq}^{(Q)}(x,t)= \frac{1}{(2\pi)^{n-1}2\pi i} \int_{E_{n-1}} e^{i(x',\zeta)}\,d\zeta \int_{\sigma-i\infty}^{\sigma+i\infty} e^{pt}\widetilde G_{iq}(\zeta,p,x_n)\, \frac{dp}{(p+\zeta^{2b})^{Q}}, \]
which possess the property
\[ \left[\frac{\partial}{\partial t}+(-1)^b\Delta_{x'}^b\right]^Q G_{iq}^{(Q)}=G_{iq}. \]
Application of the above-mentioned methods of estimating the integral (3) makes it possible to show that for \(t<0\), \(G_{iq}^{(Q)}=0\), while for \(t>0\) the inequality holds
\[ \left|D_t^{\mu}D_x^{\nu}G_{iq}^{(Q)}\right| \le C_1 t^{-\frac{n-1+2b-\sigma_q-t_j+\nu+2b\mu-2bQ}{2b}} \exp\left(-C_2 \frac{|x|^{\frac{2b}{2b-1}}} {t^{\frac{1}{2b-1}}}\right), \]
which, in turn, permits one to obtain sharp estimates of the solution of problem (1) in various norms.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
8 IV 1964
CITED LITERATURE
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V. A. Solonnikov, DAN, 157, No. 1 (1964).
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T. Ya. Zagorskii, Mixed problems for systems of partial differential equations of parabolic type, Lvov, 1961.
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S. D. Eidelman, DAN, 142, No. 4 (1962).
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S. D. Eidelman, DAN, 149, No. 4 (1963).
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V. P. Mikhailov, DAN, 126, No. 6 (1959).
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Ya. B. Lopatinskii, Scientific Notes of the Lvov Polytechnic Institute, vol. 38, phys.-math. ser., No. 2 (1956).
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V. A. Solonnikov, Izv. AN SSSR, ser. matem., 128, No. 3 (1964).
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S. Agmon, A. Douglis, L. Nirenberg, Comm. Pure and Appl. Math., 12, No. 4 (1959).