Complex powers of differential operators on manifolds with conical singularities

We construct the complex powersA _z for an elliptic cone (or Fuchs type) differential operatorA on a manifold with boundary. We show thatA _z exists as an entire family ofb-pseudodifferential operators. We also examine the analytic structure of the Schwartz kernel ofA _z , both on and off the diagonal. Finally, we study the meromorphic behavior of the zeta function Tr(A _z ). Supported by a Ford Foundation Fellowship administered by the National Research Council.     

Metric homology for isolated conical singularities

The paper is devoted to the calculation of Local Metric Homology for spaces with isolated conical singularities. We show that in this case the result is similar to the corresponding result for Intersection Homology. The main tool is a parametric version of the following theorem of Federer and Fleming: small cycles are trivial.     

Bounded imaginary powers of differential operators on manifolds with conical singularities

 We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L _p -spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers A ^z , zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Received: 12 June 2001; in final form: 3 June 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 35J70, 47A10, 35K57     

The fractional porous medium equation on manifolds with conical singularities II

This is the second of a series of two papers that studies the fractional porous medium equation, ${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma }{(|u|}^{m-1}u)=0$ with $m>0$ and $\sigma \in (0,1]$, posed on a Riemannian manifold with isolated conical singularities. The first aim of the article is to derive some useful properties for the Mellin–Sobolev spaces including the Rellich–Kondrachov theorem and Sobolev–Poincaré, Nash and Super Poincaré type inequalities. The second part of the article is devoted to the study the Markovian extensions of the conical Laplacian operator and its fractional powers. Then based on the obtained results, we establish existence and uniqueness of a global strong solution for $L_{\infty }-$initial data and all $m>0$. We further investigate a number of properties of the solutions, including comparison principle, $L_p-$contraction and conservation of mass. Our approach is quite general and thus is applicable to a variety of similar problems on manifolds with more general singularities.     

On the completeness of the generalized eigenfunctions of elliptic operators on manifolds with conical singularities

We show the completeness of the system of generalized eigenfunctions of closed extensions of elliptic cone operators under suitable conditions on the symbols (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence and maximal Lp-regularity of solutions for the porous medium equation on manifolds with conical singularities

We consider the porous medium equation on manifolds with conical singularities and show existence, uniqueness, and maximal L^p-regularity of a short-time solution. In particular, we obtain information on the short time asymptotics of the solution near the conical point. Our method is based on bounded imaginary powers results for cone differential operators on Mellin–Sobolev spaces and R-sectoriality perturbation techniques.     

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.     

Elliptic translators on manifolds with multidimensional singularities

We consider translators on manifolds with many-dimensional singularities. We state the definition of ellipticity for translators, prove a finiteness (Fredholm property) theorem, and establish an index formula.     

Elliptic translators on manifolds with point singularities

We consider translators on manifolds with singularities of the type of a transversal intersection of smooth manifolds. We give the definition of ellipticity of translators, prove the finiteness (Fredholm property) theorem, and establish an index formula for the case of point singularities.     

Spherical metrics with conical singularities on 2-spheres

Suppose that \(\theta _1,\theta _2,\ldots ,\theta _n\) are positive numbers and \(n\ge 3\). We want to know whether there exists a spherical metric on \(\mathbb {S}^2\) with n conical singularities of angles \(2\pi \theta _1,2\pi \theta _2,\ldots ,2\pi \theta _n\). A sufficient condition was obtained by Mondello and Panov (Int Math Res Not 2016(16):4937–4995, 2016). We show that their condition is also necessary when we assume that \(\theta _1,\theta _2,\ldots ,\theta _n \not \in \mathbb {N}\).     

The Arnold conjecture for a product of monotone manifolds and Calabi-Yau manifolds

We prove the Arnold conjecture for a product of finitely many monotone symplectic manifolds and Calabi-Yau manifolds. The key point of our proof is realized by suitably choosing perturbations of the almost complex structures and Hamiltonian functions for the product case. Supported by the National Natural Science Foundation of China     

Prescribing gaussian curvatures on surfaces with conical singularities

We consider prescribing Gaussian curvatures on surfaces with conical singularities. In a critical case, we obtain the best constant in an inequality. Then, by using the “distribution of mass” analysis, we are able to provide some sufficient conditions for a function to be the Gaussian curvature of some pointwise conformai metric.