Elliptic Integro-differential Operators Research Articles

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on subsolutions of the equation − A v + c ( x ) v = 0 , x ∈ D , where D ⊂ R d is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered “quantitative” versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

Uniqueness for linear integro-differential equations in the real line and applications

In this work we prove the uniqueness of solutions to the nonlocal linear equation L varphi - c(x)varphi = 0 in mathbb {R}, where L is an elliptic integro-differential operator, in the presence of a positive solution or of an odd solution vanishing only at zero. As an application, we deduce the nondegeneracy of layer solutions (bounded and monotone solutions) to the semilinear problem L u = f(u) in mathbb {R} when the nonlinearity is of Allen–Cahn type. To our knowledge, this is the first work where such uniqueness and nondegeneracy results are proven in the nonlocal framework when the Caffarelli–Silvestre extension technique is not available. Our proofs are based on a nonlocal Liouville-type method developed by Hamel, Ros-Oton, Sire, and Valdinoci for nonlinear problems in dimension two.

Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation

This paper addresses saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel $K$, and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\{(x', x'') \in \mathbb{R}^m \times \mathbb{R}^m \, : \, |x'| = |x''|\}$, and vanish only in this set. We establish the uniqueness and the asymptotic behavior of the saddle-shaped solution. For this, we prove a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in "narrow" sets. The existence of the solution was already proved in part I of this work.

Mathrm {L}^p-extrapolation of non-local operators: Maximal regularity of elliptic integrodifferential operators with measurable coefficients

The aim of this article is to deepen the understanding of the derivation of mathrm {L}^p-estimates of non-local operators. We review the mathrm {L}^p-extrapolation theorem of Shen (2005) which builds on a real variable argument of Caffarelli and Peral (1998) and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear, for example, in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire (2015). As an application, we prove resolvent estimates and maximal regularity properties in mathrm {L}^p-spaces of non-local elliptic integrodifferential operators.

Semilinear integro-differential equations, I: Odd solutions with respect to the Simons cone

This is the first of two papers concerning saddle-shaped solutions to the semilinear equation LKu=f(u) in R2m, where LK is a linear elliptic integro-differential operator and f is of Allen-Cahn type.Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone {(x′,x″)∈Rm×Rm:|x′|=|x″|}, and vanish only on this set. By the odd symmetry, LK coincides with a new operator LKO which acts on functions defined only on one side of the Simons cone, {|x′|>|x″|}, and that vanish on it. This operator LKO, which corresponds to reflect a function oddly and then apply LK, has a kernel on {|x′|>|x″|} which is different from K.In this first paper, we characterize the kernels K for which the new kernel is positive and therefore one can develop a theory on the saddle-shaped solution. The necessary and sufficient condition for this turns out to be that K is radially symmetric and τ↦K(τ) is a strictly convex function.Assuming this, we prove an energy estimate for doubly radial odd minimizers and the existence of saddle-shaped solution. In a subsequent article, part II, further qualitative properties of saddle-shaped solutions will be established, such as their asymptotic behavior, a maximum principle for the linearized operator, and their uniqueness.

Dirichlet problems with discontinuous coefficients and Feller semigroups

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. Intuitively, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary. Our approach is based on real analysis techniques.

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.

Solution to HJB Equations with an Elliptic Integro-Differential Operator and Gradient Constraint

The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton-Jacobi-Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled $d$-dimensional L\'evy process.

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.

On the Existence of Feller Semigroups with Discontinuous Coefficients

This paper is devoted to the functional analytic approach to the problem of the existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. In other words, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary.

Fast Numerical Solution of Parabolic Integrodifferential Equations with Applications in Finance

We numerically solve parabolic problems $u_t+\cA u=0$ in $(0,T)\times\Omega$, $T < \infty$, where $\Omega\subset\mathbb{R}$ is a bounded interval and $\cA$ is a strongly elliptic integrodifferential operator of order $\rho\in[0,2]$. A discontinuous Galerkin (dG) discretization in time and a wavelet discretization in space are used. The densely populated matrices in the corresponding linear systems of equations are replaced by sparse ones using appropriate wavelet compression techniques. The linear systems in each time step are solved by an incomplete GMRES iteration. Under these conditions, we show that the complexity of our algorithm is linear (up to some logarithmic terms) in the number of spatial degrees of freedom and present error estimates. Applications to purely discontinuous Lévy processes arising in finance are given.

Boundary value problems for Waldenfels operators

We study boundary value problems for second-order elliptic integro-differential operators. We prove existence and uniqueness in the framework of Bessel potential spaces.

Invariant Measures and Symmetry Property of Lévy Type Operators

Second order elliptic integro-differential operators (Levy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.

A degenerate Neumann problem for quasilinear elliptic integro-differential operators

This paper is devoted to the study of the following degenerate Neumann problem for a quasilinear elliptic integro-differential operator \(\) Here \({\mathcal W}\) is a second-order elliptic integro-differential operator of Waldenfels type and \({\mathcal L}u = a(x) \frac{\partial u}{\partial{\boldsymbol\nu}}+b(x)u\) is a first-order Ventcel' operator with a(x) and b(x) being non-negative smooth functions on \(\Gamma\) such that \(a(x) +b(x)>0\) on \(\Gamma\). Classical existence and uniqueness results in the framework of Holder spaces are derived under suitable regularity and structure conditions on the nonlinear term f(x,u,Du).

Boundary value problems for elliptic integro-differential operators

Boundary value problems for elliptic integro-differential operators

Boundary value problems for elliptic integro-differential operators

Boundary value problems for elliptic integro-differential operators