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Title Variable Branching Asymptotics on Manifolds with Edge (joint work with A. Volpato)

Name Bert-Wolfgang Schulze
Country Germany
Email schulze@math.uni-potsdam.de
Submit Time 2010-02-28 05:03:37
Special Session 35: Geometric and Singular Analysis
Solutions to elliptic equations (degenerate in a typical way) on a manifold with edge are expected to have an asymptotic behaviour in the distance variable $r$ to the edge, with complex exponents and logarithmic powers, depending on the variable $y$ on the edge. Similarly as for conical singularities the coefficients of the asymptotics also depend on the variables $x$ on the cross section of the local cones. Even for smooth solutions off the edge the dependence with respect to $x,y$ can be very complicated. In general, after applying the Mellin transform in $r$ we obtain families of meromorphic functions with poles depending on $y$ of variable multiplicity (i.e., of a branching behaviour). The position of poles is determined by the exponents in the asymptotics and the multiplicities by the exponents of the logarithmic terms (plus $1$). If solutions are not smooth, the Sobolev smoothness $s$ of coefficients of the asymptotics may also depend on $y$, with a corresponding variable and branching behaviour. We give a general answer in terms of elliptic regularity in edge spaces with such asymptotics by means of parametrices within the edge calculus, in fact, a refinement of the calculus with continuous asymptotics.