Related papers: Degenerate operators on the half-line

Examples are given of degenerate elliptic operators on smooth, compact manifolds that are not globally regular in $C^\infty$. These operators degenerate only in a rather mild fashion. Certain weak regularity results are proved, and an…

We study parameter-dependent operators on a manifold with edge and construct new classes of elliptic elements in the corner calculus on an infinite cone with a singular base

We consider the degenerate elliptic operator acting on $C^2$ functions on $[0,\infty)^d$: \[ L f(x)=\sum_{i=1}^d a_i(x) x_i^{\alpha_i} \frac{\partial^2 f}{\partial x_i^2} (x) +\sum_{i=1}^d b_i(x) \frac{\partial f}{\partial x_i}(x), \] where…

We study a class of linear parabolic equations in divergence form with degenerate coefficients on the upper half space. Specifically, the equations are considered in $(-\infty, T) \times \mathbb{R}^d_+$, where $\mathbb{R}^d_+ = \{x \in…

In this paper, we study the following singular nonlinear elliptic problem \begin{equation}\label{eq:1} \left\{ \begin{array}{ll} \displaystyle (-\Delta)^{\frac \alpha 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}}\quad &{\rm in…

This paper continues the program that was initiated in \cite{Dav18} and continued in \cite{DSVG24}, where a high-dimensional limiting technique was developed and used to prove certain parabolic theorems from their elliptic counterparts. The…

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving the Hardy-Leray potentials \begin{equation}\label{0} -\Delta u+\frac{\mu}{|x|^2} u=u^p\quad {\rm in}\quad \Omega\setminus\{0\},\qquad…

A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}<\delta_{x_j}, \cdot>\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a…

In this paper we introduce some fully nonlinear second order operators defined as weighted partial sums of the eigenvalues of the Hessian matrix, arising in geometrical contexts, with the aim to extend maximum principles and removable…

For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and…

We study the vertical and conical square functions defined via elliptic operators in divergence form. In general, vertical and conical square functions are equivalent operators just in $L^2$. But when this square functions are defined…

We study operators on a singular manifold, here of conical or edge type, and develop a new general approach of representing asymptotics of solutions to elliptic equations close to the singularities. The idea is to construct so-called…

This paper deals with regularity theory for two-phase free boundary variational problems ruled by degenerate elliptic operators.

We consider a class of elliptic and parabolic problems, featuring a specific nonlocal operator of fractional-laplacian type, where integration is taken on variable domains. Both elliptic and parabolic problems are proved to be uniquely…

In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very…

We study Cauchy problems associated to elliptic operators acting on vector-valued functions and coupled up to the first-order. We prove pointwise estimates for the spatial derivatives of the semigroup associated to these problems in the…

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operators $-\mathop{\operatorname{div}} A \nabla$ by first and zero order terms, whose coefficients lie in critical…

We establish an optimal C^{1,\alpha}-regularity for viscosity solutions of degenerate/singular fully nonlinear elliptic equations by finding minimal regularity requirements on the associated operator.

We investigate a connection between solvability of the Dirichlet problem for an infinitely degenerate elliptic operator and the validity of an Orlicz-Sobolev inequality in the associated subunit metric space. For subelliptic operators it is…

We consider a class of fully nonlinear nonlocal degenerate elliptic operators which are modeled on the fractional Laplacian and converge to the truncated Laplacians. We investigate the validity of (strong) maximum and minimum principles,…