相关论文: Extension problem and Harnack's inequality for som…

In this paper, we consider the obstacle problem for the fractional Laplace operator $(-\Delta)^s$ in the Euclidian space $\mathbb{R}^n$ in the case where $1<s<2$. As first observed in \cite{Y}, the problem can be extended to the upper…

We introduce a simple new method, based on the Caffarelli-Silvestre extension and a Duhamel-type formula, to derive exact pointwise identities for fractional commutators and nonlinear compositions associated with the fractional Laplacian on…

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic PDEs in $N$ dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last $N-1$…

In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical…

In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional $g$-Laplacian operator and lower order terms by appealing to sub- and supersolution methods. Moreover, we also state the existence of…

We develop physically admissible lattice models in the harmonic approximation which define by Hamilton's variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…

We present a way of defining the Dirichlet-to-Neumann operator on general Hilbert spaces using a pair of operators for which each one's adjoint is formally the negative of the other. In particular, we define an abstract analogue of trace…

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The…

This paper presents a new formulation of the fractional Laplacian operator $(-\Delta)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-\Delta)^s$ as a composition of the classical Laplace differential operator…

Under the lack of variational structure and nondegeneracy, we investigate three notions of \textit{generalized principal eigenvalue} for a general infinity Laplacian operator with gradient and homogeneous term. A Harnack inequality and…

In this paper we consider approximations of Neumann problems for the integral fractional Laplacian by continuous, piecewise linear finite elements. We analyze the weak formulation of such problems, including their well-posedness and…

We obtain improved fractional differentiability of solutions to the Banach-space valued Finsler $\gamma$-Laplacian defined on a $\sigma$-convex, $\tau$-smooth Banach space. The operators we consider are non-linear and very degenerately…

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set $D$. This yields a unique representation of such functions as integrals against measures on $D^c\cup…

In this paper, we focus on a space-time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak…

We investigate the existence of extremals for Hardy-Sobolev inequalities involving the Dirichlet fractional Laplacian of order s, 0<s<1, on half-spaces.

The paper is concerned with a posteriori estimates for approximations of boundary value problems generated by the spectral fractional Laplace operator. The derivation is based upon the Stinga--Torrea extension, which generalizes the…

We consider a quasilinear equation involving the $n-$Laplacian and an exponential nonlinearity, a problem that includes the celebrated Liouville equation in the plane as a special case. For a non-compact sequence of solutions it is known…

For the fractional Laplace equation, a surprising observation is the non-uniqueness for the basic Dirichlet type problems. In this paper, a somewhat sharp uniqueness condition for the fractional Laplace equation is established. We derive…

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack…