MATHEMATICS

M. I. VISHIK, G. I. ESKIN

SINGULAR ELLIPTIC EQUATIONS AND SYSTEMS OF VARIABLE ORDER

(Presented by Academician I. G. Petrovskii, 11 I 1964)

1. Singular elliptic operators (s.e.o.) of variable order on a closed manifold. Let an equation be given on a closed manifold \(M^n\)

\[ L_{\alpha(x)}u \equiv K_{\alpha(x)}u + Tu = f(x), \qquad x \in M^n, \tag{1} \]

where the operator \(K_{\alpha(x)}\) in a neighborhood \(V_j \subset M^n\) has the form

\[ K_{\alpha(x)}\varphi_j u \equiv \int_{R^n} K_{\alpha(x)}(x, x-y)\varphi_j(y)u(y)\,dy + T_1\varphi_j u, \qquad x \in V_j; \tag{2} \]

\(\operatorname{supp}\varphi_j \subset V_j\), \(\varphi_j \in C^\infty\). It is assumed that the symbol of the operator \(K_{\alpha(x)}\), i.e. the Fourier transform \(K_{\alpha(x)}(x,z)\) with respect to \(z\): \(F_z K_{\alpha(x)}(x,z)=\widetilde K_{\alpha(x)}(x,\xi)\), is a homogeneous function of \(\xi\) of order \(\alpha(x)\), so that \(\widetilde K_{\alpha(x)}(x,t\xi)=t^{\alpha(x)}\widetilde K_{\alpha(x)}(x,\xi)\). For simplicity let \(\widetilde K_{\alpha(x)}(x,\xi)\) be infinitely differentiable with respect to \(x\) and \(\xi\) (\(\xi \ne 0\)). The operators \(T\) and \(T_1\) in (1) and (2) are subordinate. We shall say that \(K_{\alpha(x)}\) is a singular elliptic operator (s.e.o.) if \(\widetilde K_{\alpha(x)}(x,\xi)\ne 0\), \(x \in M^n\), \(\xi \ne 0\).

2. Spaces of functions of variable order of smoothness. We introduce the space \(H^{\beta(x)}(M^n)\) with norm \(\|u\|_{\beta(x)}=\|S_{\beta(x)}u\|_0+\|u\|_{\beta_0}\), where \(S_{\beta(x)}\) is an s.e.o. with symbol \((1+|\xi|)^{\beta(x)}\), \(\beta(x)\in C^\infty(M^n)\), \(\beta_0=\min \beta(x)-1\), and \(\|u\|_l\) is the usual norm of order \(l\).

Theorem 1. 1) If \(\beta_1(x)<\beta_2(x)\), then \(H^{\beta_2(x)}\subset H^{\beta_1(x)}\), and the embedding operator is completely continuous.

2) Let \(\{V_j\}\) be a covering of \(M^n\), \(\{\varphi_j\}\) the corresponding partition of unity, and let \((\beta)=\{\beta_j\}\), \((\gamma)=\{\gamma_j\}\) be such sets of numbers that \(\beta_j<\beta(x)<\gamma_j\) for \(x\in \overline V_j\).

Then \(H^{(\gamma)}\subset H^{\beta(x)}\subset H^{(\beta)}\), and the embedding operators are completely continuous; the norm in \(H^{(\delta)}\) is given by the formula \(\|u\|_{(\delta)}=\sum \|\varphi_j u\|_{\delta_j}\).

3. Normal solvability of equation (1) in \(H^{\beta(x)}\). Theorem 2. The operator \(L\) is a \(\Phi\)-operator (1) acting from the space \(H^{\beta(x)}\) into \(H^{\beta(x)-\alpha(x)}\), if \(K_{\alpha(x)}\) is an s.e.o., and \(T\) is a completely continuous operator from \(H^{\beta(x)}\) into \(H^{\beta(x)-\alpha(x)}\). The estimate holds

\[ \|u\|_{\beta(x)} \le C\left(\|f\|_{\beta(x)-\alpha(x)}+\|u\|_{\beta_0}\right). \tag{3} \]

Here and below, as a subordinate operator \(T\) it suffices to take an operator of the form \(\sum K_{\gamma_i(x)}+T_N\), where \(\widetilde K_{\gamma_i(x)}(x,\xi)=|\xi|^{\gamma_i(x)}\ln^{\delta_i}|\xi|\,\widetilde K_{0,i}(x,\xi)\); \(\gamma_i(x)<\alpha(x)\); \(\widetilde K_{0,i}(x,t\xi)=\widetilde K_{0,i}(x,\xi)\); \(T_N\) is a smoothing operator of order-

order \(N\), i.e., if \(u \in H^{\alpha_0}\), then \(T_N u \in H^{N+\alpha_0}\). \(N\) is fixed, sufficiently large, and in each concrete case its value can be specified.

4. Boundary-value problems for s.e.o. in a bounded domain. We now consider an equation of the form (1) in a bounded domain \(G \subset R^n\) with smooth boundary \(\Gamma\). As in \(\left({}^{2}\right)\), we factor the kernel \(\widetilde K_{\alpha(x)}\) on \(\Gamma\):

\[ \widetilde K_{\alpha(x)}(x,\xi)=\widetilde K_{\chi(x)}^{+}(x,\xi)/\widetilde K_{\chi(x)-\alpha(x)}^{-}(x,\xi), \]

\(\xi=(\xi',\xi_n)\), \(\xi_n\) corresponds to the normal to \(\Gamma\) at the point \(x \in \Gamma\); \(\widetilde K_{\chi(x)}^{+}(x,\xi)\ne0,\ \operatorname{Im}\xi_n \geqslant 0\), \(\widetilde K_{\chi(x)-\alpha(x)}^{-}(x,\xi)\ne0,\ \operatorname{Im}\xi_n \leqslant 0,\ \xi\ne0\). In \(\left({}^{2}\right)\) general boundary-value problems were considered for the case of constant and integral \(\chi\) and for \(\widetilde K_{\chi}^{+}\) satisfying condition c) or c′). Here the general case of nonintegral and variable \(\chi(x)\) is considered. We extend \(\widetilde K_{\chi(x)}^{+}\) to the whole space \(R^n\), preserving the ellipticity condition: \(\widetilde K_{\chi}^{+}(x,\xi)\ne0,\ x\in R^n,\ \xi\ne0\). It is proved that there exists the following representation of the operator \(L_{\alpha(x)}\) in all of \(R^n\):

\[ L_{\alpha(x)}=L^{-}\cdot L^{+}+T_N, \tag{4} \]

where \(L^{+}\) is a sum of operators with symbols \(\widetilde K_{\chi(x)}^{+}(x,\xi)\), \(\widetilde K_{\gamma_i(x)}^{+}(x,\xi)\), \(\gamma_i(x)<\chi(x)\), and all of them, for \(x\in\Gamma\), are analytic for \(\operatorname{Im}\xi_n>0\). Similarly, \(L^{-}\) is a sum of operators with symbols \([\widetilde K_{\chi(x)-\alpha(x)}^{-}(x,\xi)]^{-1}\), \(\widetilde K_{\delta_j(x)}^{-}(x,\xi)\), \(\delta_j(x)<\alpha(x)-\chi(x)\), analytic in \(\xi_n\) for \(x\in\Gamma\), \(\operatorname{Im}\xi_n<0\). By \(u(x)\) we shall denote, generally speaking, generalized functions defined in \(R^n\) and equal to zero in \(R^n\setminus(G\cup\Gamma)\). Let \(v(x)=D^{+}L^{+}u\), where \(D^{+}\) is an s.e.o. of order of homogeneity \(-M\) with symbol \(\xi_+^{-M}=(\xi_n+i|\xi'|)^{-M}\) on \(\Gamma\), \(M\) an integer, \(M\geqslant0\). By \(H_{L^{+},M}^{l(x)}(G)\) we denote the space of functions \(u(x)\) such that \(v(x)\in L_{\mathrm{loc}}^{2}(R^n)\) with finite norm

\[ \|u\|_{l(x)}=\|v(x)\|_{l(x)-\chi(x)+M}+\|u\|_{-N}, \tag{5} \]

where \(\|v\|_{\gamma(x)}=\inf\|Lv\|_{\gamma(x)}\), \(Lv\) is an extension of \(v\) from \(G\) to \(R^n\). The latter means that \(v(x)\in H^{\gamma(x)}(G)\). We note that the functions \(u(x)\in H_{L^{+},M}^{l(x)}\) have smoothness of order \(l(x)\) inside \(G\), and near \(\Gamma\), \(u(x)=O(r^{\chi(x)-M})\) for nonintegral \(\chi(x)\); \(r\) is the distance of \(x\) from \(\Gamma\). If \(L^{+}\) satisfies condition c) from \(\left({}^{2}\right)\) and, hence, \(\chi\) is integral, then \(u(x)\) may turn into a \(\delta\)-function and its derivatives on \(\Gamma\) when \(M>\chi\). Therefore, usually in the case of boundary-value problems for differential equations one takes \(M=\chi\), and then \(H_{L^{+},M}^{l}=H^{l}\) \((l(x)\equiv l)\). In the general case there is no necessity for this, and \(M\) indicates the necessary number of boundary conditions. We note that when \(M<\chi(x)\), part of the boundary conditions is contained in the fact that \(u\in H_{L^{+},M}^{l(x)}\).

On \(\Gamma\) we prescribe \(M\) boundary conditions of the form

\[ B_j u\big|_{\Gamma}=F_j(x'),\qquad x'\in\Gamma\quad (j=1,\ldots,M), \tag{6} \]

where \(B_j\) are operators of the form \(L_{\alpha(x)}\), satisfying the conditions:

1) \(B_j\cdot R^{+}=V_j+T_{j,N}\), where \(L^{+}\cdot R^{+}=I+T_N\), and for \(V_j\) condition c) from \(\left({}^{2}\right)\) holds.

2)

\[ \det\left\| \int_{\Gamma_0} B_j^{(0)}(x,\xi)\,[\widetilde K_{\chi(x)}^{+}(x,\xi)]^{-1}\, \xi_n^{\,k-1}\,d\xi_n \right\|\ne0,\qquad x\in\Gamma,\quad \xi'\ne0,\quad j,k=1,\ldots,M, \]

where \(B_j^{(0)}(x,\xi)\) is the principal part of \(\widetilde B_j(x,\xi)\), and \(\Gamma_0\) is the same contour as in \(\left({}^{2}\right)\). We note that in a number of cases condition 1) can be dropped.

Theorem 3. If \(L_{\alpha(x)}\) is an s.e.o. and \(B_j\) \((j=1,\ldots,M)\) satisfy conditions 1) and 2), then the operator corresponding to problem (1), (6) is a \(\Phi\)-operator in the corresponding spaces, and the estimate holds

\[ \|u\|_{l(x)}\leqslant C\left( \|f\|_{l(x)-\alpha(x)} + \sum_{j=1}^{M}\|F_j\|_{l(x)-m_j(x)-1/2} + \|u\|_{l(x)-1} \right), \tag{7} \]

where \(f\in H^{l(x)-\alpha(x)}(G)\), \(F_j\in H^{l(x)-m_j(x)-1/2}(\Gamma)\), \(m_j(x)=\operatorname{ord}\widetilde B_j\), \(l(x)>\max(m_j(x)+1/2,\alpha(x),\varkappa(x)-M)\).

This theorem, in the case \(\widetilde K^+\) satisfying condition c) and \(M=\varkappa\), coincides with Theorem 3 in \((^2)\).

5. Problems with additional potentials.
In the case of an integer negative \(M\), in the norm of the space \(H_{l+,M}^{\chi(x)}\), instead of the boundary conditions (6), one should add to equation (1) \(|M|\) terms of potential type (see \((^2)\)). Thus, the equation considered is

\[ L_{\alpha(x)}\left(u(x)+\sum_{k=1}^{|M|}G_k g_k(x')\right)=f(x),\qquad x\in G,\quad x'\in\Gamma, \tag{8} \]

where \(G_k\) are operators of the type \(L_{\alpha(x)}\) (see \((^2)\)), satisfying the following conditions:

\(1')\) \(L^+G_k=W_k+T_{k,N}\), where \(W_k\) satisfies condition c) from \((^2)\).

\(2')\) Since \(\widetilde W_k^{(0)}\) satisfies condition c), on \(\Gamma\) \(\xi_+^M\Pi^+\widetilde W_k^{(0)}(x,\xi)=P_k(x,\xi',\xi_n)+R_k^+(x,\xi)\), where \(\widetilde W_k^{(0)}\) is the principal part of \(\widetilde W_k\), \(\widetilde W_k^{(0)}=\Pi^+\widetilde W_k^{(0)}+\widetilde W_k^-\), \(\Pi^+\widetilde W_k^{(0)}\) is analytic for \(\operatorname{Im}\xi_n>0\) and decreases as \(\xi_n\to\infty\), \(P_k(x,\xi',\xi_n)\) are polynomials in \(\xi_n\) of degree not higher than \(|M|-1\), \(|R_k^+(x,\xi)|\le C(x,\xi')/(|\xi_n|+|\xi'|)\).

Condition \(2')\) on \(G_k\) consists in the fact that the polynomials \(P_k\) \((k=1,\ldots,|M|)\) are linearly independent.

Theorem 4. The operator corresponding to equation (8), where \(L_{\alpha(x)}\) is a properly elliptic operator and \(G_k\) satisfy conditions \(1')\), \(2')\), is a \(\Phi\)-operator in the corresponding spaces; moreover, the estimate

\[ \|u\|_{l(x)}+\sum_{k=1}^{|M|}\|g_k\|_{l(x)+\alpha_k(x)+1/2} \le C\left(\|f\|_{l(x)-\alpha(x)}+\|u'\|_{l(x)-1} +\sum_{k=1}^{|M|}\|g_k\|_{\delta_k(x)}\right) \tag{9} \]

\[ \alpha_k(x)=\operatorname{ord}G_k,\qquad \delta_k(x)=l(x)+\alpha_k(x)-1/2. \]

Let us note that the solution \(u(x)\) of equation (8) has smoothness \(\varkappa(x)+|M|\) up to the boundary \(\Gamma\). This is achieved by isolating terms of the form of potentials \(G_k g_k(x')\), which contain the principal singularities of the solution of the equation \(L_{\alpha(x)}v=f\) in \(G\).

Remark. a) We note that the potentials in (8) may be taken in the same form as in \((^2)\), i.e., the left-hand side of (8) may be replaced by the following:
\(L_{\alpha(x)}u+\sum \widehat G_k g_k\). b) The number of potentials may be increased to the number \(|M|+s\), adding at the same time additionally \(s\) boundary conditions of the form (6).

Example. We restrict ourselves to the simplest example. Let, for simplicity, in (1) \(T=0\) and let \(K_{\alpha(x)}\) be the operator with symbol \(|\xi|^\alpha\) (\(\alpha\) constant), \(G\) a bounded domain. Then for \(\alpha<0\), \(K_\alpha\) is an integral operator. For \(\alpha>0\), \(K_\alpha\) is an integro-differential operator. \(\widetilde L^+=\xi_+^{\alpha/2}\), so that \(\varkappa=\alpha/2\). Let \(\alpha>0\). Take \(M=[\alpha/2]\) and denote \(\gamma=\alpha/2-M\). Then as boundary operators \(B_j\) one may take operators with symbols \(\widetilde B_j=\xi_+^{\gamma+j-1}\) \((j=1,\ldots,M)\), and all the conditions of Theorem 3 will be fulfilled. If \(\alpha<0\), we take, for example, \(M\) such that \(\alpha/2=-M+\gamma\), \(0\le\gamma<1\). As \(G_k\) in (8) it is sufficient to take kernels with symbols \(\xi_+^{M-1-k-\gamma}\) \((k=1,\ldots,|M|)\), in order that all the conditions of Theorem 4 be fulfilled.

6. Systems of properly elliptic operators. Consider a system of equations of the form (1) in the domain \(G\) with symbol \(\widetilde K_{\alpha(x)}(x,\xi)\), which is a matrix of order \(r\times r\). The ellipticity condition now consists in the fact that \(\det \widetilde K_{\alpha(x)}(x,\xi)\ne0\), \(\xi\ne0\), \(x\in G\cup\Gamma\), while the elements \(\widetilde K_{\alpha(x)}(x,\xi)\) are homogeneous functions in \(\xi\) of order \(\alpha(x)\). We shall call the system properly elliptic at points \(x\in\Gamma\) if \(\widetilde K_{\alpha(x)}(x,\xi)\) admits a factorization of the form (see \((^3)\))

\[ \widetilde K_{\alpha(x)}(x,\xi)=\widetilde K_-^{-1}(x,\xi)S^{-1}S_+\widetilde K_+(x,\xi), \tag{10} \]

where \(\widetilde K_+\), \(\widetilde K_-\) are homogeneous matrices of order zero with determinant not vanishing for \(\operatorname{Im}\xi_n \geqslant 0\), \(\operatorname{Im}\xi_n \leqslant 0\) \((\xi \ne 0)\), respectively; \(S_+=\|\delta_{jk}\xi_+^{\varkappa_j(x)}\|\), \(S_-=\|\delta_{jk}\xi_+^{\varkappa_j(x)-\alpha(x)}\|\), \(\widetilde K_+\), \(\widetilde K_-\) depend smoothly on \(x\) and \(\xi\) \((\xi \ne 0)\). Let \(L^+\), \(L^-\), \(R^+\) be operator matrices defined in the same way as in the scalar case, i.e. \(L_{\alpha(x)}=L^- L^+ + T_N\), \(x\in R^n\), \(L^+R^+=I+T_N\). Let, further, \(M=(M_1,\ldots,M_r)\) be an arbitrary integer vector, and, for definiteness, \(M_i>0\), \(1\leqslant i\leqslant k\), \(M_j\leqslant 0\), \(k+1\leqslant j\leqslant r\). Denote
\[ M_+=\sum_{i=1}^{k}M_i,\qquad M_-=\sum_{j=k+1}^{r}|M_j|. \]
Introduce the space \(H^l_{L^+,M}(G)\) with norm
\[ |||u|||_{l(x)}=\sum_{k=1}^{r}\|v_k\|_{\gamma_k(x)}, \]
where \(v=(v_1,\ldots,v_r)\),
\[ v=D^+L^+u,\qquad \widetilde D^+=\|\xi_+^{-M_i}\delta_{ik}\|,\qquad \gamma_k(x)=l(x)-\varkappa_k(x)+M_k; \]
\(\|\ \|_{\gamma_k(x)}\) is the norm in \(H^{\gamma_k(x)}(G)\) (see item 4).

The boundary-value problem for (1) in this case is posed as follows:
\[ L_{\alpha(x)}(u+G_1g)=f(x),\qquad x\in G, \tag{11} \]
\[ B(u+G_1g)\big|_{\Gamma}=F(x'),\qquad x'\in\Gamma, \tag{12} \]
where \(G_1\) is a matrix of order \(r\times M_-\), and the matrix \(B\) is of order \(M_+\times r\). We formulate conditions on \(B\) and on \(G_1\):

1) \(B\cdot R^+=V+T_N\), where \(V\) satisfies condition c).

2) Let \(V_0\) be the principal part of \(V\). Denote by \(P(x,\xi',\xi_n)\) the vector-polynomial \((P_1,\ldots,P_r)\) such that \(P_j\equiv 0\) for \(j\geqslant k+1\), while for \(1\leqslant j\leqslant k\), \(P_j\) is an arbitrary polynomial in \(\xi_n\) of degree \(M_j-1\) with coefficients depending on \(\xi'\) and \(x\). Let \(Z\) be a matrix of order \(r\times M_+\) with entries equal to zero or to \(\xi_n^i\), such that \(P=ZC\), where \(C\) is an \(M_+\)-dimensional vector with components equal to the coefficients of the polynomials \(P_j\). It is required that
\[ \det\int_{\Gamma_0}\widetilde V_0(x,\xi)Z\,d\xi_n\ne 0,\qquad \xi'\ne 0,\quad x\in\Gamma. \]

\(1')\) \(G_1=R^+W+T_N\), where the matrix \(W=\|W_{jk}\|\) satisfies condition c).

\(2')\) Let \(W_{jk}^{(0)}\) be the principal part of \(W_{jk}\). We have:
\[ \xi_+^{-M_j}\Pi^+W_{jk}^{(0)}(x,\xi)=P_{jk}(x,\xi)+R_{jk}^+(x,\xi), \]
where
\[ |R_{jk}^+(x,\xi)|\leqslant C(x,\xi')/(|\xi_n|+|\xi'|), \]
\(P_{jk}(x,\xi)\) are polynomials in \(\xi_n\), and \(P_{jk}\equiv 0\) for \(1\leqslant j\leqslant k\), while \(\operatorname{ord}P_{jk}\leqslant |M_j|-1\) for \(k+1\leqslant j\leqslant r\). Condition \(2')\) on \(G_1\) consists in the fact that the vector-polynomials
\[ P_l=(P_{k+1,l},\ldots,P_{r,l}) \]
are linearly independent \((l=1,\ldots,M_-)\).

Theorem 5. If conditions 1), 2), \(1'\), \(2'\) are fulfilled, then problem (11), (12) for a system properly elliptic on \(\Gamma\) is normally solvable in the corresponding spaces, and the estimate
\[ |||u|||_{l(x)}+\sum_{k=1}^{M_-}\|g_k\|_{l(x)+\alpha_k(x)+1/2} \]
\[ \leqslant C\left(\|f\|_{l(x)-\alpha(x)} +\sum_{k=1}^{M_+}\|F_k\|_{l(x)-m_{\tilde k}(x)-1/2} +|||u|||_{l(x)-1} +\sum_{k=1}^{M_-}\|g_k\|_{l(x)+\alpha_k(x)-1/2}\right), \]
where \(B=\|B_{ik}\|\), \(\operatorname{ord}\widetilde B_{ik}=m_i(x)\), \(G_1=\|G_{1,ik}\|\), \(\operatorname{ord}\widetilde G_{1,ik}=\alpha_k(x)\), \(u\in H^l_{L^+,M}\), \(f\in H^{\,l(x)-\alpha(x)}(G)\), \(F_k\in H^{\,l(x)-m_k(x)-1/2}(\Gamma)\), \(g_k\in H^{\,l(x)+\alpha_k(x)+1/2}(\Gamma)\).

Received
11 I 1964

CITED LITERATURE

¹ I. Ts. Gokhberg, M. G. Krein, UMN, 12, issue 2 (74), 43 (1957). ² M. I. Vishik, G. I. Eskin, DAN, 155, No. 1 (1964). ³ I. Ts. Gokhberg, M. G. Krein, UMN, 13, issue 2, 3 (1958).