Nonlinear Integro-differential Operators Research Articles

Electron kinetics in a high-Z plasmoid

The problem of the electron dynamics on a closed magnetic field line passing through a high- $Z$ plasmoid is considered. The electron kinetic equation is integrated over bounce motion and pitch angle, reducing the independent variables to a single adiabatic invariant plus time. Integration of the full Landau self-collision operator is carried out exactly, resulting in a nonlinear integro-differential operator in the new invariant. Conservation laws and the $H$ theorem of the integrated self-collision operator are proven. Numerical solutions of the integrated kinetic equation are obtained with a self-consistent quasineutral electric potential, given the initial condition of a cold plasmoid immersed in a hot ambient plasma. The fact that cold electrons are deeply trapped in a potential with a parabolic peak leads to exactly 3/4 the usual rate of collisional heating by the ambient plasma, independent of any other parameters.

The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group

We study the obstacle problem related to a wide class of nonlinear integro-differential operators, whose model is the fractional subLaplacian in the Heisenberg group. We prove both the existence and uniqueness of the solution, and that solutions inherit regularity properties of the obstacle such as boundedness, continuity and Hölder continuity up to the boundary. We also prove some independent properties of weak supersolutions to the class of problems we are dealing with. Armed with the aforementioned results, we finally investigate the Perron–Wiener–Brelot generalized solution by proving its existence for very general boundary data.

Regularity estimates for fractional orthotropic p-Laplacians of mixed order

Abstract We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Hölder estimate.

Regularity results of nonlinear perturbed stable-like operators

We consider a class of fully nonlinear integro-differential operators where the nonlocal integral has two components: the non-degenerate one corresponds to the $\alpha$-stable operator and the second one (possibly degenerate) corresponds to a class of lower order Lévy measures. Such operators do not have a global scaling property. We establish Hölder regularity, Harnack inequality and boundary Harnack property of solutions of these operators.

Principal eigenvalues of a class of nonlinear integro-differential operators

We consider a class of nonlinear integro-differential operators and prove existence of two principal (half) eigenvalues in bounded smooth domains with exterior Dirichlet condition. We then establish simplicity of the principal eigenfunctions in viscosity sense, maximum principles, continuity property of the principal eigenvalues with respect to domains etc. We also prove an anti-maximum principle and study existence result for some nonlinear problem via Rabinowitz bifurcation-type results.

Regularity for fully nonlinear integro-differential operators with kernels of variable orders

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in Caffarelli and Silvestre (2009). Since the order of differentiability of the kernel is not characterized by a single number, we use the constant Cφ=∫Rn1−cosy1|y|nφ(|y|)dy−1instead of 2−σ, where φ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and Hölder estimates of viscosity solutions to the nonlinear integro-differential equations.

Obstacle problems for integro-differential operators: regularity of solutions and free boundaries

We study the obstacle problem for integro-differential operators of order $2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove the following dichotomy at all free boundary points $x_0\in\partial\{u=\varphi\}$: (i) either $u(x)-\varphi(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+\alpha})$ for some $c>0$, (ii) or $u(x)-\varphi(x)=o(|x-x_0|^{1+s+\alpha})$, where $d$ is the distance to the contact set $\{u=\varphi\}$. Moreover, we show that the set of free boundary points $x_0$ satisfying (i) is open, and that the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near those points. These results were only known for the fractional Laplacian \cite{CSS}, and are completely new for more general integro-differential operators. The methods we develop here are purely nonlocal, and do not rely on any monotonicity-type formula for the operator. Thanks to this, our techniques can be applied in the much more general context of fully nonlinear integro-differential operators: we establish similar regularity results for obstacle problems with convex operators.

Hölder continuity up to the boundary for a class of fractional obstacle problems

We deal with the obstacle problem for a class of nonlinear integrodifferential operators, whose model is the fractional p-Laplacian with measurable coefficients. In accordance with well-known results for the analog for the pure fractional Laplacian operator, the corresponding solutions inherit regularity properties from the obstacle, both in the case of boundedness, continuity, and Holder continuity, up to the boundary.

The obstacle problem for nonlinear integro-differential operators

We investigate the obstacle problem for a class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator with measurable coefficients. Amongst other results, we will prove both the existence and uniqueness of the solutions to the obstacle problem, and that these solutions inherit regularity properties, such as boundedness, continuity and Holder continuity (up to the boundary), from the obstacle.

Neumann homogenization via integro-differential operators

In this note we describe how the Neumann homogenization of fully non- linear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is mo- tivated by a new integro-differential representation for nonlinear operators with a com- parison principle which we also introduce. In the simple case that the original domain is an infinite strip with almost periodic Neumann data, this leads to an almost peri- odic homogenization problem involving a fully nonlinear integro-differential operator on the Neumann boundary. This method gives a new proof- which was left as an open question in the earlier work of Barles- Da Lio- Lions- Souganidis (2008)- of the result obtained recently by Choi-Kim-Lee (2013), and we anticipate that it will generalize to other contexts.

Regularity for Fully Nonlinear Integro-differential Operators with Regularly Varying Kernels

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre (Comm. Pure Appl. Math. 62, 597–638, 2009) are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Holder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels K σ,β satisfying $$K_{\sigma,\beta}(y)\asymp\frac{ 2-\sigma}{|y|^{n+\sigma}}\left( \log\frac{2}{|y|^{2}}\right)^{\beta(2-\sigma)} \text{near zero} $$ with respect to σ ∈ (0, 2) close to 2 (for a given $\beta \in \mathbb R$ ), where the regularity estimates do not blow up as the order σ ∈ (0, 2) tends to 2.

Regularity Results for Fully Nonlinear Integro-Differential Operators with Nonsymmetric Positive Kernels: Subcritical Case

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the operator $\sigma$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a H\"older regularity, and an interior $\rm C^{1,\alpha}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $\sigma$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.

Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff–Backelman–Pucci estimate corresponding to the full class \({\mathcal {S}^{\mathfrak {L}_0}}\) of uniformly elliptic nonlinear equations with 1 < σ < 2 (subcritical case) and to their subclass \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) with 0 < σ ≤ 1. We show that \({\mathcal {S}_{\eta}^{\mathfrak {L}_0}}\) still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and C 1,α-regularity of the solutions by obtaining decay estimates of their level sets in each cases.

Multipoint problem for intehrodyferentsialnyh partial differential equations, not solvable on higher derivatives

The correctness of a problem with multi–point conditions on temporary variable of linear partial differential equations not solved as to the highest derivative with respect to time, perturbed by the nonlinear integro-differential operator is investigated

Invariant relation in the problem of size quantization in a self-consistent potential near the semiconductor boundary

The invariant relation in the eigenvalue problem is found for a one-dimensional nonlinear integrodifferential operator acting on the normalized eigenfunctions. The invariance of the obtained relationship follows, in particular, from the fact that it is independent of the detailed form of the operator. This problem arises in the physics of a two-dimensional electron system near the semiconductor boundary, where the potential well is formed, on the one hand, by a high potential barrier at the boundary and, on the other, by the intrinsic electric field screened by two-dimensional electrons filling the well. The invariant relates the energy of the size-quantization level to the average size of electron wave function in the well.

Nonlinear mathematical theory of perforated viscoelastic thin plates with its applications

In this paper, the nonlinear mathematical theory of perforated viscoelastic thin plates, by the Kármán’s hypotheses of plates with large deflection and the Boltzmann’s constitutive law of linear viscoelastic materials, is established. One could see that the governing equations, boundary conditions and constraining conditions generally are of nonlinear integro-differential operator types and that they further generalize the mathematical theory for perforated elastic thin plates and could be also reduced to the existing mathematical theory of viscoelastic thin plates without holes. As an application, the nonlinear dynamical stability of a viscoelastic annular plate is analyzed and the effect of parameters on the stability is considered by using Galerkin averaging method and numerical methods in nonlinear dynamics. Some helpful conclusions are obtained. Specially, a new method calculating the Lyapunov exponent spectrum of dynamical systems excited periodically is suggested. By using this method, the computation time can be greatly reduced.