Elliptic equations on manifolds with cusp-type singularities.

*(English. Russian original)*Zbl 0959.58029
Dokl. Math. 58, No. 2, 242-244 (1998); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 362, No. 4, 453-455 (1998).

From the text: This paper studies elliptic differential equations on manifolds having cusp-type point singularities. More precisely, assume that we are given a compact topological space \(M\) and a system of points \(\{m_1,m_2, \dots,m_N\}\) on this manifold such that

(i) \(M\subset \{m_1, m_2, \dots, m_N\}\) has the structure of a \(C^\infty\)-manifold;

(ii) in a neighborhood of each point \(m_j\), \(j=1,2, \dots,N\), the homeomorphism \[ \alpha_j: \bigl([0,1) \times\Omega \bigr)/ \bigl(\{0\} \times\Omega \bigr)\to U_j \tag{1} \] of the cone \(K=([0,1) \times\Omega)/ (\{0\}\times \Omega)\) over a smooth compact manifold \(\Omega\) with no edge onto some neighborhood \(U_j\) of the point \(m_j\) in \(M\) is given; the coordinates corresponding to representation (1) are denoted hereafter by \((r,\omega)\);

(iii) the restriction of \(\alpha_j\) to \(K\setminus \{0\}\) is a diffeomorphism;

(iv) the structural ring of differential operators in a neighborhood of the point \(\alpha_j\) consists of the operators of the form \[ \widehat H=r^{-(k_j+1)\mu} \sum^\mu_{j= 0} \widehat A_j(r) \left(-ir^{k_j +1}{\partial \over\partial r} \right)^j. \tag{2} \] Here, \(\mu\) denotes the order of the operator \(\widehat H\), \(\widehat A_j (r)\) are differential operators on \(\Omega\) that have smooth coefficients and depend smoothly on the variable \(r\) up to \(r=0\) (of course, \(\mu\) and \(\widehat A_j(r)\) may depend on \(\widehat H)\), and the number \(k_j\) (dependent only on the point \(m_j)\) is an integer, called the degree of this point.

If conditions (i)–(iv) are fulfilled, \(m_j\) is called a cusp point of degree \(k_j\).

The goal of this study is to introduce the concept of the ellipticity of differential operators on a manifold with cusp-type singularities, to establish corresponding finiteness theorems, and to describe the asymptotic behavior of elements of the kernels of elliptic operators in the vicinity of singular points of the manifold \(M\).

(i) \(M\subset \{m_1, m_2, \dots, m_N\}\) has the structure of a \(C^\infty\)-manifold;

(ii) in a neighborhood of each point \(m_j\), \(j=1,2, \dots,N\), the homeomorphism \[ \alpha_j: \bigl([0,1) \times\Omega \bigr)/ \bigl(\{0\} \times\Omega \bigr)\to U_j \tag{1} \] of the cone \(K=([0,1) \times\Omega)/ (\{0\}\times \Omega)\) over a smooth compact manifold \(\Omega\) with no edge onto some neighborhood \(U_j\) of the point \(m_j\) in \(M\) is given; the coordinates corresponding to representation (1) are denoted hereafter by \((r,\omega)\);

(iii) the restriction of \(\alpha_j\) to \(K\setminus \{0\}\) is a diffeomorphism;

(iv) the structural ring of differential operators in a neighborhood of the point \(\alpha_j\) consists of the operators of the form \[ \widehat H=r^{-(k_j+1)\mu} \sum^\mu_{j= 0} \widehat A_j(r) \left(-ir^{k_j +1}{\partial \over\partial r} \right)^j. \tag{2} \] Here, \(\mu\) denotes the order of the operator \(\widehat H\), \(\widehat A_j (r)\) are differential operators on \(\Omega\) that have smooth coefficients and depend smoothly on the variable \(r\) up to \(r=0\) (of course, \(\mu\) and \(\widehat A_j(r)\) may depend on \(\widehat H)\), and the number \(k_j\) (dependent only on the point \(m_j)\) is an integer, called the degree of this point.

If conditions (i)–(iv) are fulfilled, \(m_j\) is called a cusp point of degree \(k_j\).

The goal of this study is to introduce the concept of the ellipticity of differential operators on a manifold with cusp-type singularities, to establish corresponding finiteness theorems, and to describe the asymptotic behavior of elements of the kernels of elliptic operators in the vicinity of singular points of the manifold \(M\).

##### MSC:

58J05 | Elliptic equations on manifolds, general theory |

58J32 | Boundary value problems on manifolds |

35J40 | Boundary value problems for higher-order elliptic equations |

58J42 | Noncommutative global analysis, noncommutative residues |