On rays of minimal growth for elliptic cone operators

Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains

We study the problem of the existence and nonexistence of positive solutions to a superlinear second-order divergence type elliptic equation with measurable coefficients $(*)$: $-\nabla\cdot a\cdot\nabla u=u^p$ in an unbounded cone--like domain $G\subset\bf R^N$ $(N\ge 3)$. We prove that the critical exponent $p^*(a,G)=\inf\{p>1 : (*) \hbox{has a positive supersolution in} G\}$ for a nontrivial cone-like domain is always in $(1,N/(N-2))$ and in contrast with exterior domains depends both on the geometry of the domain $G$ and the coefficients $a$ of the equation.

Positive solutions to singular semilinear elliptic equations with critical potential on cone-like domains

We study the existence and nonexistence of positive (super-)solutions to a singular semilinear elliptic equation $$-\nabla\cdot(|x|^A\nabla u)-B|x|^{A-2}u=C|x|^{A-σ}u^p$$ in cone--like domains of $\R^N$ ($N\ge 2$), for the full range of parameters $A,B,σ,p\in\R$ and $C>0$. We provide a complete characterization of the set of $(p,σ)\in\R^2$ such that the equation has no positive (super-)solutions, depending on the values of $A,B$ and the principle Dirichlet eigenvalue of the cross--section of the cone. The proofs are based on the explicit construction of appropriate barriers and involve the ana…

Resolvents of elliptic boundary problems on conic manifolds

This paper is a continuation of the investigation of resolvents of elliptic operators on conic manifolds from math.AP/0410178 and math.AP/0410176 to the case of manifolds with boundary and realizations of operators under boundary conditions. We prove the existence of sectors of minimal growth for realizations of boundary value problems for cone operators under natural ellipticity conditions on the symbols. Special attention is devoted to the clarification of the analytic structure of the resolvent.

图上椭圆算子的锥流形视角

A conic manifold perspective of elliptic operators on graphs

本文基于椭圆锥算子理论，为图上具有 Coulomb 型奇异势的二阶正则奇异微分算子提供了存在最小增长扇区的简单且显式的充分条件。

锥伪微分算子的预解式、渐近展开及应用

Resolvents of cone pseudodifferential operators, asymptotic expansions and applications

本文研究了带边紧流形上加权 Sobolev 空间上椭圆锥伪微分算子预解式的结构与渐近行为，得出谱参数趋于无穷时的预解式渐近展开，并推导出热迹、zeta 函数展开及解析指标公式。