Asymptotics of the parabolic Green function for an elliptic operator on a manifold with conical points

Abstract

ABSTRACT. We construct the asymptotics as t ---* 0 of the trace of the operator exp(-tP) for an elliptic operator P on a manifold with conical points.KEY WORDS: elliptic operator,-heat kernel, trace asymptotics, manifold with conical points, Green's function. w IntroductionLet P be a self-adjoint positive definite elliptic operator of order 21 in the space L2 on a closed compact manifold M of dimension m. For the case in which the coefficients of the operator are smooth, the following statement is well known (e.g., see [1] H(t, x, x) ..~ ~ ck(z)t (-'~+2k)/(2') , t --* +0.( 1) k=0This azymptotics provides important information on the operator P and the geometry of the manifold (see ill).If the coefficients of the operator are discontinuous at some point, then the expansion becomes nonuniform and the coefficients ck(x) axe singular at the point of discontinuity. We consider an elliptic operator on a manifold M with a conical point x0. This means (e.g., see [21)that M\{x0} is an m-dimensional C ~ manifold and there exists a neighborhood U of the point x0 and a mapping k of this neighborhood (a diffeomorphism preserving the conical structure) onto a neighborhood of the vertex 0 of a conical manifold X. We denote by S a fixed directing submartifold of X (S is a compact manifold of dimension m -1 intersecting each ray from X at one point). For the points of X and hence for the points in the neighborhood U we can introduce the polar coordinates x = r, w, where r E R+, w E S and r-ix E S.To simplify the notation, we make no distinction between a function u on U and the function u o k -1 on X. We also choose a positive density dx on M smooth on M \ {x0 } and assume that dx = r m-1 dr dw in the neighborhood U, where dw is a smooth positive density on S. Of course, the manifold can have any number of conical points; in what follows, we consider only one such point x0.An important example of an operator on a manifold with a conical point is an operator on a closed smooth manifold with symbol having an isolated singularity at the point. At this point, the principal symbol must have the limit that can depend on the direction of approach, and lower terms can have power-law singularities. In this case, X = R "~ and S = S '~-] .The operator is defined in the space L2(M) of Cd-valued functions via the quadratic form (Pu, u)= fM ~ (aat,a2(x)D~tu, D'~u) dx, aat,a2(x)=a*2,at(x), D~j = -iO~j ,

Heat Kernel Asymptotics of Zaremba Boundary Value Problem

“…For such a treatment the reader is referred to the papers [38,39,29,28,11,32] for the smooth case and to [26,17,18,16,14,15,41,37,35,33,27] for the singular case.…”

“…Let us stress here that we are not going to provide a rigorous construction of the parametrix with all the estimates, which, for a singular boundary-value problem, is a task that would require a separate paper. For such a treatment the reader is referred to the papers [38,39,29,28,11,32] for the smooth case and to [26,17,18,16,14,15,41,37,35,33,27] for the singular case.…”

“…manifolds with singularities (corners, edges, cones etc.). There is a large body of literature on this subject where the problem is studied from an abstract function-analytical point of view [26,17,18,16,14,15,41,37,35,33,27]. However, the study of heat kernel asymptotics of Zaremba type problems is quite new, and there are only some preliminary results in this area [5,21,20].…”

The heat kernel on curvilinear polygonal domains in surfaces

“…There is substantial work in the literature on heat trace expansions in the settings (1) and (2). For conical singularities with no boundary on the link, a non-exhaustive list of works concerning the heat kernel and its trace is: [43], [22], [26], [27], [12], [13], [38]. In the case (2) of vertices which locally have straight edges, a non-exhaustive list includes [45], [39], [21], and [24].…”

Full asymptotic expansion of the heat trace for non–self–adjoint elliptic cone operators

“…However, for certain classes of self-adjoint operators on some singular manifolds there are many results concerning resolvent, heat kernel and heat trace asymptotics; in this context we want to refer to [4], [5], [7], [8], [9], [12], [26], [28] and [35]. In particular, Lesch [28] generalized the method introduced by Brüning and Seeley in [4] and obtained asymptotic expansions of the heat trace for self-adjoint differential operators with coefficients that are independent of the radial variable near the singularities.…”