Higher Order Fractional Laplacian Research Articles

Oscillatory phenomena for higher-order fractional Laplacians

Oscillatory phenomena for higher-order fractional Laplacians

On a super polyharmonic property of a higher-order fractional Laplacian

On a super polyharmonic property of a higher-order fractional Laplacian

The s-polyharmonic extension problem and higher-order fractional Laplacians

We provide a detailed description of the relationships between the fractional Laplacian of order 2s∈(0,n) on Rn and the s-polyharmonic extension operator.

Super poly-harmonic properties, Liouville theorems and classification of nonnegative solutions to equations involving higher-order fractional Laplacians

In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlinearities $f(x,u,Du,\cdots)$ including conformally invariant and odd order cases. In particular, our results completely improve the classification results for third order equations in Dai and Qin \cite{DQ1} by removing the assumptions on integrability. We also derive a characterization for $\alpha$-harmonic functions via averages in the appendix.

Brake orbits and heteroclinic connections for first order mean field games

In this paper, we are concerned with equations \eqref{PDE} involving higher-order fractional Laplacians. By introducing a new approach, we prove the super poly-harmonic properties for nonnegative solutions to \eqref{PDE} (Theorem \ref{Thm0}). Our theorem seems to be the first result on this problem. As a consequence, we derive many important applications of the super poly-harmonic properties. For instance, we establish Liouville theorems, integral representation formula and classification results for nonnegative solutions to fractional higher-order equations \eqref{PDE} with general nonlinearities $f(x,u,Du,\cdots)$ including conformally invariant and odd order cases. In particular, our results completely improve the classification results for third order equations in Dai and Qin \cite{DQ1} by removing the assumptions on integrability. We also derive a characterization for $\alpha$-harmonic functions via averages in the appendix.

A Liouville theorem for the higher-order fractional Laplacian

We study Navier problems involving the higher-order fractional Laplacians. We first obtain nonexistence of positive solutions, known as the Liouville-type theorems, in the upper half-space [Formula: see text] by studying an equivalent integral form of the fractional equation. Then we show symmetry for positive solutions on [Formula: see text] through a delicate iteration between lower-order differential/pseudo-differential equations split from the higher-order equation.

On the loss of maximum principles for higher-order fractional Laplacians

We study existence, regularity, and qualitative properties of solutions to linear problems involving higher-order fractional Laplacians $(-\Delta)^s$ for any $s>1$. Using the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $s\in(n,n+1)$ with $n\in\mathbb N$ odd; moreover, using a representation formula for solutions, we derive regularity and positivity preserving properties whenever the domain is the whole space or a ball. In the case of the whole space we analyze the Riesz kernel, which provides a fundamental solution, while in the case of the ball we show the validity of Boggio's representation formula for all integer and fractional powers of the Laplacian $s>0$. Our proofs rely on characterizations of $s$-harmonic functions using higher-order Martin kernels, on a decomposition of Boggio's formula, and on elliptic regularity theory.

Green function and Martin kernel for higher-order fractional Laplacians in balls

We give the explicit formulas for the Green function and the Martin kernel for all integer and fractional powers of the Laplacian s>1 in balls. As consequences, we deduce interior and boundary regularity estimates for solutions to linear problems and positivity preserving properties. Our proofs rely on a characterization of suitable s-harmonic functions and on a differential recurrence equation.

Integro-differential systems of mixed type involving higher order fractional Laplacian

ABSTRACTIn this paper, we deal with a new class of non-local operators that we term integro-differential systems of mixed type. We study the behaviour of solutions of this system when the diffusion term involves higher order fractional powers of the Laplacian. Moreover, we prove that the solution of the system decays faster than a power with an exponent given by the smallest index of the fractional power of the Laplacian.

Nonexistence Results for Nonlocal Equations with Critical and Supercritical Nonlinearities

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| n+σ is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case a ij ≡ 0 and for σ = 2 in case that (a ij ) is a positive definite symmetric matrix. Our nonexistence results concern Dirichlet problems for L in star-shaped domains with critical and supercritical nonlinearities (where the criticality condition is in relation to n and σ). We also establish nonexistence of bounded solutions to semilinear equations involving other nonlocal operators such as the higher order fractional Laplacian (− Δ) s (here s > 1) or the fractional p-Laplacian. All these nonexistence results follow from a general variational inequality in the spirit of a classical identity by Pucci and Serrin.

Classification of positive solutions for an elliptic system with a higher-order fractional Laplacian

Classification of positive solutions for an elliptic system with a higher-order fractional Laplacian