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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 $$M$$.
##### 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