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MATHEMATICS
V. P. MIKHAILOV
ON THE FIRST BOUNDARY VALUE PROBLEM FOR SOME SEMIBOUNDED OPERATORS
(Presented by Academician I. G. Petrovskii, 1 II 1963)
I. Let
\[ A(x,\xi)=\sum_{\alpha} a_{\alpha}(x)\xi^{\alpha} \tag{1} \]
be a polynomial in \(\xi=(\xi_1,\ldots,\xi_n)\) with real coefficients \(a_{\alpha}(x)\), depending on the parameter \(x=(x_1,\ldots,x_n)\in \overline Q\), where \(Q\) is some domain of \(n\)-dimensional \(x\)-space, \(\alpha=(\alpha_1,\ldots,\alpha_n)\), \(\xi^\alpha=\xi_1^{\alpha_1}\cdots \xi_n^{\alpha_n}\).
Consider the set \(E(x)\) of integer points \(\alpha\) lying in the region \(\alpha_1\geqslant 0,\ldots,\alpha_n\geqslant 0\) and such that \(a_{\alpha}(x)\ne 0\). Consider also the minimal convex polyhedron \(G(x)\) containing \(E(x)\). Denote the union of \(E(x)\) and \(G(x)\) over all \(x\in \overline Q\) by \(E\) and \(G\), respectively. Let \(e_1,\ldots,e_N\), \(e_i=(e_i^1,\ldots,e_i^n)\), \(i=1,\ldots,N\), be the set of vertices of \(G\), and let \(\Gamma_1,\ldots,\Gamma_M\) be the set of its \((n-1)\)-dimensional faces.
Consider the face \(\Gamma_k\) with exterior normal \(m_k=(m_k^1,\ldots,m_k^n)\), \(k=1,\ldots,M\). Let
\[ A_{\Gamma_k}(\xi)=\sum_{\alpha\in\Gamma_k} a_{\alpha}(x)(i\xi)^\alpha. \]
Since \(\alpha\in\Gamma_k\), \(A_{\Gamma_k}(\xi)\) is a generalized homogeneous polynomial in \(\xi\).* Take a vector \(c=(c_1,\ldots,c_n)\), whose components \(c_r\), \(r=1,\ldots,n\), are equal either to \(+1\) or to \(-1\) (there are \(2^n\) such vectors in all), and make the change of variables
\[ \xi_r=c_r \zeta_r^{m_k^r},\quad r=1,\ldots,n, \]
where \(-\infty<\zeta_r<\infty\), \(r=1,\ldots,n\) (we shall assume, without loss of generality, that all coordinates of the normal \(m_k\) are integers).
Under such a change of variables, \(A_{\Gamma_k}(\xi)\) becomes a homogeneous polynomial \(\widetilde A_{\Gamma_k}^{(c)}(\zeta)\) (the superscript \(c\) indicates that a change of variables has been made with some vector \(c\)). Represent \(\widetilde A_{\Gamma_k}^{(c)}(\zeta)\) in the form
\[ \widetilde A_{\Gamma_k}^{(c)}(\zeta) = \zeta_1^{\rho_1}\cdots \zeta_n^{\rho_n}\, \widetilde{\widetilde A}_{\Gamma_k}^{(c)}(\zeta), \]
where \(\rho_1,\ldots,\rho_n\) are certain integers \(\rho_i\geqslant 0\), \(i=1,\ldots,n\), and \(\widetilde{\widetilde A}_{\Gamma_k}^{(c)}(\zeta)\) is a polynomial homogeneous in \(\zeta\), whose terms have no common factors of the form \(\zeta_r^{\theta_r}\) with \(\theta_r>0\), \(r=1,\ldots,n\).
Definition 1. A face \(\Gamma_k\), \(k=1,\ldots,M\), will be called principal if its (exterior) normal \(m_k\) makes an acute angle with at least one of the coordinate axes \(Oa_s\), \(s=1,\ldots,n\).
Definition 2. A face \(\Gamma_k\), \(k=1,\ldots,M\), will be called leading if the homogeneous polynomials \(\widetilde{\widetilde A}_{\Gamma_k}^{(c)}(\zeta)\), for all choices of \(c\), do not vanish on the unit sphere \(|\zeta|=1\).
Definition 3. A vertex \(e_p\), \(p=1,\ldots,N\), will be called principal if it belongs to at least one principal face.
Let the principal faces of \(G\) be the faces \(\Gamma_1,\ldots,\Gamma_{M_1}\), \(M_1\leqslant M\), and the principal vertices be the vertices \(e_1,\ldots,e_{N_1}\), \(N_1\leqslant N\).
* For brevity of exposition, it is everywhere assumed in the note that \((m_k,i_s)\ne 0\), \(k=1,\ldots,M\), \(s=1,\ldots,n\), where \(i_s\) is the direction vector of the axis \(Oa_s\).
Assumptions.
-
We shall assume that the intersection \(E_0\) of all \(E(x)\), \(x\in \overline Q\), is nonempty and that all principal vertices \(e_1,\ldots,e_{N_1}\) of \(G\) lie in \(E_0\).
-
For all \(x\in \overline Q\),
\[ \lim_{\xi\to\infty} A(x,\xi)=\infty \tag{2} \]
for real \(\xi\).
- All principal faces are leading.
Theorem 1. There exists a constant \(\gamma_0>0\) such that for all \(x\in \overline Q\) and \(\xi\to\infty\),
\[ \gamma_0^{-1}\leq \frac{A(x,\xi)}{\xi^{e_1}+\cdots+\xi^{e_N}} \leq \gamma_0 . \]
II. Consider the differential operator
\[ H(x,D)=\sum_{\alpha} h_\alpha(x)D^\alpha \tag{3} \]
\(x=(x_1,\ldots,x_n)\in \overline Q\), where \(Q\) is some bounded domain of the \(n\)-dimensional space \(x\), \(D^\alpha=D_1^{\alpha_1}\cdots D_n^{\alpha_n}\), \(D_i=\partial/\partial x_i\), \(i=1,\ldots,n\); the coefficients \(h_\alpha(x)\) will be assumed to be complex-valued functions from \(C^\infty(\overline Q)\). The sets \(E(x)\), \(G(x)\), \(E,\ldots\), etc., for the operator (3) are introduced in exactly the same way as the corresponding sets in the preceding section (in the real case). We shall assume that, for \(H(x,D)\), assumptions 1 and 3 are satisfied, made with respect to the operator \(A(x,-iD)\). Then the following holds.
Theorem 2. In order that the operator (3) be hypoelliptic, it is necessary and sufficient that:
1) for real \(\xi\to\infty\),
\[
H(x,i\xi)\to\infty \quad \text{for } x\in \overline Q;
\]
2) on each coordinate axis \(O\alpha_s\), \(s=1,\ldots,n\), of the space \(\{\alpha\}\) there must lie a principal vertex of the polyhedron \(G\);
3) the (outer) normals of the principal faces of \(G\) must have only positive coordinates.
This theorem is proved with the aid of Hörmander’s results \((^1)\) for hypoelliptic equations with variable coefficients and Theorem 1. Let us note that the necessity of the second condition of Theorem 2 was established in the paper \((^2)\).
III. Consider, in a domain \(Q\) of the \(n\)-dimensional space \(x=(x_1,\ldots,x_n)\), the linear differential equation
\[ P_1(x,D)u\equiv \sum_{\alpha} c_\alpha(x)D^\alpha u=f(x) \tag{4} \]
with sufficiently smooth complex coefficients \(c_\alpha(x)\). Suppose that the characteristic polynomial for the operator \(P_1(x,D)\) has the form
\[ P_1(x,i\xi)=\operatorname{Re} P_1(x,i\xi)+i\,\operatorname{Im} P_1(x,i\xi) \equiv A(x,\xi)+iB(x,\xi), \]
\[ A(x,\xi)=\sum_{\alpha} a_\alpha(x)\xi^\alpha,\qquad B(x,\xi)=\sum_{\alpha} b_\alpha(x)\xi^\alpha, \tag{5} \]
where \(\xi=(\xi_1,\ldots,\xi_n)\) is a real vector, and the polynomial \(A(x,\xi)\) is the polynomial from Section I.
In addition to the polyhedron \(G\) constructed for the polynomial \(A(x,\xi)\), consider also the polyhedron
\[
G_1=\left(\bigcup_{\alpha\in G} R_\alpha\right)\cup G,
\]
where \(R_\alpha\) is the set of points \(\beta=(\beta_1,\ldots,\beta_n)\) for which \(0\leq \beta_i\leq \alpha_i\), \(i=1,\ldots,n\). Obviously, \(G_1\) is a convex polyhedron, with \(n\) of its faces (not principal ones) lying on the coordinate planes. Denote by \(f_1,\ldots,f_{N_2}\) the principal vertices of the polyhedron \(G_1\); it turns out that all numbers \(f_k^1,\ldots,f_k^n\), \(k=1,\ldots,N_2\) \((f_k=(f_k^1,\ldots,f_k^n))\), are even.
Lemma 1. There exists \(\lambda_0\geq 0\) such that the quadratic norm
\[ [u,u]=\bigl((A(x,D)+\lambda_0)u,u\bigr) \tag{6} \]
one may take for the square of the norm in \(C_0^\infty(Q)\) (the parentheses \((\, , \,)\) denote the scalar product in \(\mathscr L_2(Q)\)).
In the Hilbert space \(\mathscr H_A(Q)\), obtained by completing \(C_0^\infty(Q)\) in the metric (6), the scalar product may also be introduced in the following way:
\[ [u,v]_1=\int_Q \sum_{i=1}^{N_2} D^{f_i/2}u\,D^{f_i/2}v\,dx . \]
Functions from \(\mathscr H_A(Q)\) satisfy on the boundary \(S\) of the domain \(Q\) certain homogeneous boundary conditions. For brevity, we describe these conditions for the case when \(S\) contains no planar pieces parallel to the coordinate planes.
Lemma 2. If \(u\in \mathscr H_A(Q)\), then
\[ D^\beta u\big|_S=0 \tag{7} \]
for any integer vector \(\beta\) which satisfies at least one of the inequalities \(f_r-\beta>0,\ r=1,\ldots,N_2\) (the vector \(\beta\geq 0\) if all its coordinates are nonnegative; the vector \(\beta>0\) if all its coordinates are nonnegative and at least one coordinate is strictly positive). Equality (7) is understood in the mean.
Lemma 3. Let every point \(\alpha\) corresponding to \(b_\alpha(x)\ne 0\) in \(Q\) (see (5)) either be an interior point of \(G_1\) or, being a boundary point, not lie on any of the principal faces of \(G_1\). Then there exist constants \(\lambda_1\geq 0\) and \(\lambda_2>0\) such that
\[ \lambda_2^{-1}[u,u]\leq |(P_1(x,D)u+\lambda_1u,u)|\leq \lambda_2[u,u] \tag{8} \]
for all \(u\in C_0^\infty(Q)\).
In the case when the coefficients \(c_\alpha(x)\) of the operator \(P(x,D)\) are real, a stronger statement holds.
Lemma 3′. There exist constants \(\lambda_1\geq 0\) and \(\lambda_2>0\) such that inequalities (8) hold for any \(u\in C_0^\infty(Q)\), if the \(c_\alpha(x)\) in (4) are real and every \(\alpha\) for which \(c_\alpha(x)\ne 0\) in \(Q\) satisfies one of the following conditions:
a) \(\alpha\) lies inside \(G_1\);
b) \(\alpha\), being on the boundary of \(G_1\), does not lie on any of its principal faces;
c) \(\alpha\) lies on the boundary of \(G_1\) or even outside \(G_1\), but \(\alpha-i_k\) (if \(\alpha-i_k\geq 0\)) for all \(k=1,\ldots,n\) lie either inside \(G_1\) or on the boundary of \(G_1\), but not on a principal face (here \(i_k=(0,\ldots,0,1,0,\ldots,0)\)).
These lemmas are analogous to the corresponding lemma in \((^3)\) and are an extension to the operators considered here of Gårding’s well-known lemma \((^4)\) for elliptic operators.
Definition. A function \(u(x)\in \mathscr H_A(Q)\) will be called a generalized solution of the first boundary-value problem for equation (4), if the integral identity
\[ (u,P^*(x,D)v)=(f,v) \tag{9} \]
(\(f\in \mathscr L_2(Q)\), \(P^*(x,D)\) is the operator formally adjoint to the operator \(P(x,D)\)) holds for every function \(v\in C_0^\infty(Q)\).
Theorem 3. The problem of finding a generalized solution of the first boundary-value problem for equation (4) is Fredholm if the conditions of Lemma 3 (or 3′) are fulfilled. For the operator \(P(x,D)+\lambda\), with \(\lambda\) sufficiently large, the first boundary-value problem is uniquely solvable.
IV. We carry out in the \((n+1)\)-dimensional space \(\{a\}=\{a_0,a_1,\ldots,a_n\}\) the following construction: each point of the boundary \(\Delta_1\) of the convex polyhedron \(G_1\), lying in the plane \(a_0=0\) (\(G_1\) was constructed in Sec. III), is joined to the point \((2\nu_0+1,0,\ldots,0)\), where \(\nu_0\) is an integer \(\nu_0\geq 0\) such that
\[ 2\nu_0+1<\sigma=\max_{1\leq i\leq n}\ \sup_{(0,a_1,\ldots,a_n)\in G_1} a_i \]
(we assume, for definiteness—
that \(\sup \alpha_i=\sigma\) for \(i=1,\ldots,i_0\) and \(\sup \alpha_i<\sigma\) for \(i>i_0\). The conical surface \(\Delta_2\) constructed in this way (taken together with \(\Delta_1\)) will be taken as the lateral surface of the pyramid \(G_2\), whose lower base will be taken to be \(G_1\). The pyramid \(G_2\) is a convex polyhedron in the space \(\{\alpha\}\). We shall denote its principal faces by \(\Delta_2^i\), \(i=1,\ldots,M_1\). Analogously one constructs the polyhedron \(G_3\) with the same base as \(G_2\), but with vertex \((2\nu_0,0,\ldots,0)\). Suppose that in the domain \(Q_2\), situated in the \((n+1)\)-dimensional space \(\{x\}=\{x_0,x_1,\ldots,x_n\}\) and bounded by the closed surface \(S_2\), the differential equation
\[ P_2(x,D)u \equiv \sum_{\alpha\in\Delta_2} r_\alpha(x)D^\alpha u+\sum_{\alpha\in G_3} r_\alpha(x)D^\alpha u=f(x) \tag{10} \]
is given, with real (for simplicity) sufficiently smooth coefficients. Suppose that
\[ r_{2\nu_0+1,\,0,\ldots,0}(x)\ne 0 \quad \text{for } x\in \overline{Q}_2; \tag{11} \]
\[ \operatorname{Re}\sum_{\alpha\in\Delta_2} r_\alpha(x)(i\xi)^\alpha \ge \gamma^2 \sum_{k=1}^{N_1} |\xi|^{\widetilde e_k} \quad \text{for } x\in \overline{Q}_2,\ \gamma>0, \tag{12} \]
where \(\widetilde e_k\) are the vertices of the minimal convex polyhedra lying on \(\Delta_2^i\), \(i=1,\ldots,M_1\), and containing all points \(\alpha\in\Delta_2\) such that \(r_\alpha(x)\ne0\) for \(x\in \overline Q\).
Suppose also that \(S_2\) has only two points \(B=(x_0^b,x_1^b,\ldots,x_n^b)\) and \(H=(x_0^H,\ldots,x_n^H)\) at which the tangent plane to \(S_2\) is perpendicular to the axis \(Ox_0\). Denote by \(\mathscr H_{P_2}(Q_2)\) the space obtained by completing \(C_0^\infty(Q_2)\) in the metric
\[ \{u,u\}=\int_{Q_2}\left[(D_0^{\nu_0}u)^2+\sum (D^{\widetilde e_i/2}u)^2\right]\,dx. \]
By a generalized solution in \(\mathscr H_{P_2}(Q_2)\) of equation (10) we shall mean (for \(f\in L_2(Q_2)\)) a function \(u(x)\in \mathscr H_{P_2}(Q_2)\) for which
\[ (u,P_2^*(x,D)v)=(f,v) \]
is fulfilled for every \(v\in C_0^\infty(Q_2)\) (\(P_2^*\) is the operator formally adjoint to \(P_2\)).
Theorem 4. The problem of solving equation (10) in \(\mathscr H_{P_2}(Q_2)\) is a Fredholm-type problem if the domain \(Q_2\) lies inside one of the cylinders
\[ |x_i-x_i^H|=c\,(x_0-x_0^H)^{\frac{2\nu_0+1}{\sigma}+\varepsilon}, \quad i=1,\ldots,i_0;\ x_0\ge x_0^H, \]
and inside one of the cylinders
\[ |x_i-x_i^b|=c\,(x_0^b-x_0)^{\frac{2\nu_0+1}{\sigma}+\varepsilon}, \quad i=1,\ldots,i_0;\ x_0\le x_0^b, \]
for some \(\varepsilon>0\).
In §§ III and IV problems with homogeneous boundary conditions were considered. The case of nonhomogeneous conditions is considered analogously to the corresponding case in (3) or (5).
Received
29 I 1963
REFERENCES
\(^{1}\) L. Hörmander, Comm. pure and appl. math., 11, No. 2, 197 (1958).
\(^{2}\) F. Trèves, Ann. Inst. Fourier, 9, 1 (1959).
\(^{3}\) V. P. Mikhailov, DAN, 149, No. 6 (1963).
\(^{4}\) Gårding, Math. Scand., 1, No. 1, 55 (1953).
\(^{5}\) S. M. Nikol’skii, DAN, 146, No. 4, 767 (1962).