Related papers: Fractional De Giorgi classes and applications to n…
We derive quantitatively the Harnack inequalities for kinetic integro-differential equations. This implies H\"older continuity. Our method is based on trajectories and exploits a term arising due to the non-locality in the energy estimate.…
This work is devoted to the study of the existence of at least one weak solution to nonlocal equations involving a general integro-differential operator of fractional type. As a special case, we derive an existence theorem for the…
The notion of a local fractional derivative (LFD) was introduced recently for functions of a single variable. LFD was shown to be useful in studying fractional differentiability properties of fractal and multifractal functions. It was…
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new…
The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…
In this note we treat the equations of fractional elasticity. After establishing well-posedness, we show a compactness result related to the theory of homogenization. For this, a previous result in (abstract) homogenization theory of…
The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem $(P)$ (See below) is addressed. Secondly, we investigate the question of $H^s$ versus $C^0$- weighted minimizers of the functional…
A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…
This paper defines a new class of fractional differential operators alongside a family of random variables whose density functions solve fractional differential equations equipped with these operators. These equations can be further used to…
In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^{\gamma}$-regularity for…
This book presents a comprehensive regularity theory for solutions of elliptic, parabolic, and kinetic equations. The foundation of this theory was laid by E. De Giorgi's groundbreaking resolution of Hilbert's nineteenth problem in 1956.…
This article concerns with the global H\"older regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$.…
We present a general framework for the rigorous numerical analysis of time-fractional nonlinear parabolic partial differential equations, with a fractional derivative of order $\alpha\in(0,1)$ in time. The framework relies on three…
In this work, we study the fractional power series solutions around regular singular point x=0 of conformable fractional Bessel differential equation and fractional Bessel functions. Then, we compare fractional solutions with ordinary…
In this paper we study the solutions of different forms of fractional equations on the unit sphere $\mathbb{S}_{1}^{2}$ $\subset \mathbb{R}^{3}$ possessing the structure of time-dependent random fields. We study the correlation functions of…
We consider mixed local and nonlocal quasilinear parabolic equations of $p$-Laplace type and discuss several regularity properties of weak solutions for such equations. More precisely, we establish local boundeness of weak subsolutions,…
Singular perturbations have been used to select solutions of (non-convex) variational problems with a multiplicity of minimizers. The prototype of such an approach is the gradient theory of phase transitions by L. Modica, who specialized…
We study the regularity of solutions of elliptic fractional systems of order 2s, $s \in (0, 1)$, where the right hand side f depends on a nonlocal gradient and has the same scaling properties as the nonlocal operator. Under some structural…
We discuss some basic properties of the eigenfunctions of a class of nonlocal operators whose model is the fractional p-Laplacian.
We prove that certain non-local functionals defined on measurable sets Gamma-converge to the perimeter in the sense of De Giorgi.