Related papers: Asymptotic behavior for anisotropic fractional ene…
In this article we study the asymptotic behavior of anisotropic nonlocal nonstandard growth seminorms and modulars as the fractional parameter goes to 1. This gives a so-called Bourgain-Brezis-Mironescu type formula for a very general…
In this article we define a class of fractional Orlicz-Sobolev spaces on Carnot groups and, in the spirit of the celebrated results of Bourgain-Brezis-Mironescu and of Maz'ya-Shaposhnikova, we study the asymptotic behavior of the Orlicz…
Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a…
We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a…
We obtain an asymptotic representation formula for harmonic functions with respect to a linear anisotropic nonlocal operator. Furthermore we get a Bourgain-Brezis-Mironescu type limit formula for a related class of anisotropic nonlocal…
The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in $\rn$ is shown to converge to the anisotropic perimeter with respect to the moment body of $K$ as $s\to 1^-$. A corresponding result is established for…
In this paper we analyze the asymptotic behavior of several fractional eigenvalue problems by means of Gamma-convergence methods. This method allows us to treat different eigenvalue problems under a unified framework. We are able to recover…
We determine the asymptotic behaviour of (bilateral) obstacle problems for fractional energies in rather general aperiodic settings via Gamma-convergence arguments. As further developments we consider obstacles with random sizes and shapes…
We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order $2s$ when the parameter $s$ tends to zero. In particular, we show that least-energy solutions converge (up to a…
We study the asymptotic behavior of the fractional Allen--Cahn energy functional in bounded domains with prescribed Dirichlet boundary conditions. When the fractional power $s \in (0,\frac12)$, we establish establish the first-order…
Some problems in the theory and applications of stochastic processes can be reduced to solving integral equations. While explicit solutions for these equations are often elusive, valuable insights can be gained through their asymptotic…
In this paper, we consider the asymptotic behavior of the ground state solution $u_s$ of the nonlinear fractional Laplacian equation \begin{equation}\label{eq:0.1a} (-\Delta)^su+Vu=|u|^{p-2}u\quad x\in \mathbb{R}^n \end{equation} by taking…
In this paper, we study the asymptotic behavior of a semi-linear slow-fast stochastic partial differential equation with singular coefficients. Using the Poisson equation in Hilbert space, we first establish the strong convergence in the…
We investigate the asymptotic behavior in the sense of $\Gamma(L^1_{loc})$-convergence as $s\to1^-$ of anisotropic non local $s$-fractional perimeters defined with respect to general anisotropic integration kernels $k_s(\cdot)$, under the…
In this paper, we first investigate the monotonicity and limit problem of the fractional integral functions. By fixed point theorem and these new results of the fractional integral functions, we present that the Riemann-Liouville fractional…
In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen…
We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as…
In this work we study the asymptotics of the fractional Laplacian as $s\to 0^+$ on any complete Riemannian manifold $(M,g)$, both of finite and infinite volume. Surprisingly enough, when $M$ is not stochastically complete this asymptotics…
This paper studies a nonlinear shallow shell model proposed by Donnell, Vlasov, Mushtari, Galimov, and Koiter. More specifically, we address the question concerning the asymptotic behavior of minimizing solutions. Our result can be applied…
Combining the ideas of Riesz $s$-energy and $\log$-energy, we introduce the so-called $s,\log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,\log^t$-energy constants…