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Sven Jarohs

Publications and source records attributed to Sven Jarohs.

At least 19 recordsLinked to original sources
arXiv2026

$s$-harmonic functions in the small order limit

We study families $u_s$ of functions satisfying the equations $(-Δ)^s u_s=0$, $s \in (0,1)$ in a smooth bounded open set $Ω\subset \mathbb{R}^N$. The main purpose of this paper is twofold. First, we provide a detailed analysis of the asymptotics of these families in the zero order limit $s \to 0^+$. Second, we study the differentiability of $u_s$ as a function of $s$. Most of our results are devoted to the associated Poisson problem, where the family $u_s$ is determined by the exterior condition $u_s = g$ in $\mathbb{R}^N \setminus Ω$ for some fixed function $g \in L^\infty(\mathbb{R}^N \setminus Ω)$. Our results show that both the zero order asymptotics and the differentiability properties of $u_s$ can be expressed in terms of the logarithmic Laplacian of suitable extensions of $g$. This allows to deduce pointwise monotonicity properties of $u_s$ in the order parameter $s$ for a large class of functions $g$.

math.AP↗
arXiv2025

On a class of (non)local superposition operators of arbitrary order

In this paper we introduce a very general setting dealing with the superposition of operators of any positive order and provide a systematic study of them. We also provide examples and counterexamples, as well as characterizing properties of the measures and the functional spaces under consideration. Moreover, we present some applications regarding the existence theory for a class of nonlinear problems involving superposition operators of arbitrary (possibly fractional) order.

math.AP↗
arXiv2025

The Dirichlet Problem For the Logarithmic p-Laplacian

We introduce and study the logarithmic $p$-Laplacian $L_{Δ_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-Δ_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator. We present a variational framework to study the Dirichlet problems involving the $L_{Δ_p}$ in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional $p$-Laplacian and the logarithmic $p$-Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of $L_{Δ_p}$. We discuss maximum and comparison principles for $L_{Δ_p}$ in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of $L_{Δ_p}$. In addition, we prove that the first Dirichlet eigenfunction of $L_{Δ_p}$ is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian.

math.AP↗
arXiv2025

Overdetermined problems with fractional Laplacian

Let $N\geq 1$ and $s\in (0,1)$. In the present work we characterize bounded open sets $Ω$ with $ C^2$ boundary (\textit{not necessarily connected}) for which the following overdetermined problem \begin{equation*} ( -Δ)^s u = f(u) \text{ in $Ω$,} \qquad u=0 \text{ in $\mathbb{R}^N\setminus Ω$,} \qquad(\partial_η)_s u=Const. \text{ on $\partial Ω$} \end{equation*} has a nonnegative and nontrivial solution, where $η$ is the outer unit normal vectorfield along $\partialΩ$ and for $x_0\in\partialΩ$ \[ \left(\partial_η\right)_{s}u(x_{0})=-\lim_{t\to 0}\frac{u(x_{0}-tη(x_0))}{t^s}. \] Under mild assumptions on $f$, we prove that $Ω$ must be a ball. In the special case $f\equiv 1$, we obtain an extension of Serrin's result in 1971. The fact that $Ω$ is not assumed to be connected is related to the nonlocal property of the fractional Laplacian. The main ingredients in our proof are maximum principles and the method of moving planes.

math.AP↗
arXiv2025

On overdetermind problems for a general class of nonlocal operators

We study the overdetermined problem for a large family of non-local operators given by generators of subordinate Brownian motions. In particular, this family includes the fractional Laplacian, relativistic stable operators etc. We consider these problems in bounded domains, exterior domains, and in annular domains and we show that under suitable conditions, the domains and solutions are both radially symmetric. Our method uses both analytic and probabilistic tools.

math.AP↗
arXiv2025

FEM for 1D-problems involving the logarithmic Laplacian: error estimates and numerical implementation

We present the numerical analysis of a finite element method (FEM) for one-dimensional Dirichlet problems involving the logarithmic Laplacian (the pseudo-differential operator that appears as a first-order expansion of the fractional Laplacian as the exponent $s\to 0^+$). Our analysis exhibits new phenomena in this setting; in particular, using recently obtained regularity results, we prove rigorous error estimates and provide a logarithmic order of convergence in the energy norm using suitable $\log$-weighted spaces. Moreover, we show that the stiffness matrix of logarithmic problems can be obtained as the derivative of the fractional stiffness matrix evaluated at $s=0$. Lastly, we investigate the relationship between the discrete eigenvalue problem and its convergence to the continuous one.

math.NA↗
arXiv2025

Continuity of solutions to equations with weakly singular nonlocal operators

We prove boundedness and regularity estimates for weak solutions to a class of linear nonlocal equations involving integro-differential operators with almost no order of differentiability. In particular, we show that bounded weak solutions are continuous, and we provide a uniform a-priori estimates for the modulus of continuity. In contrast to earlier works, we allow the nonlocal operators to be highly anisotropic and weakly singular, and we allow the associated kernel functions to vanish close to the singularity.

math.AP↗
arXiv2024

Differentiability of the nonlocal-to-local transition in fractional Poisson problems

Let $u_s$ denote a solution of the fractional Poisson problem $$ (-Δ)^s u_s = f\quad\text{ in }Ω,\qquad u_s=0\quad \text{ on }\mathbb{R}^N\setminus Ω, $$ where $N\geq 2$ and $Ω\subset \mathbb{R}^N$ is a bounded domain of class $C^2$. We show that the solution mapping $s\mapsto u_s$ is differentiable in $L^\infty(Ω)$ at $s=1$, namely, at the nonlocal-to-local transition. Moreover, using the logarithmic Laplacian, we characterize the derivative $\partial_s u_s$ as the solution to a boundary value problem. This complements the previously known differentiability results for $s$ in the open interval $(0,1)$. Our proofs are based on an asymptotic analysis to describe the collapse of the nonlocality of the fractional Laplacian as $s$ approaches 1. We also provide a new representation of $\partial_s u_s$ for $s \in (0,1)$ which allows us to refine previously obtained Green function estimates.

math.AP↗
arXiv2023

Nonlocal operators of small order

In this work we study nonlocal operators and corresponding spaces of order strictly below one and investigate interior regularity properties of weak solutions to the associated Poisson problem depending on the regularity of the right-hand side. Our method exploits the variational structure of the problem, in particular, we prove that if the right-hand is of class $C^{\infty}$ and the kernel satisfies similar regularity properties away from its singularity, then any weak solution is of class $C^{\infty}$.

math.AP↗
arXiv2022

Nonradiality of second fractional eigenfunctions of thin annuli

In the present paper, we study properties of the second Dirichlet eigenvalue of the fractional Laplacian of annuli-like domains and the corresponding eigenfunctions. In the first part, we consider an annulus with inner radius $R$ and outer radius $R+1$. We show that for $R$ sufficiently large any corresponding second eigenfunction of this annulus is nonradial. In the second part, we investigate the second eigenvalue in domains of the form $B_1(0)\setminus \overline{B_τ(a)}$, where $a$ is in the unitary ball and $0<τ<1-|a|$. We show that this value is maximized for $a=0$, if the set $B_1(0)\setminus \overline{B_τ(0)}$ has no radial second eigenfunction. We emphasize that the first part of our paper implies that this assumption is indeed nonempty.

math.AP↗
arXiv2022

Oscillatory Phenomena for Higher-Order Fractional Laplacians

We collect some peculiarities of higher-order fractional Laplacians $(-Δ)^s$, $s>1$, with special attention to the range $s\in(1,2)$, which show their oscillatory nature. These include the failure of the polarization and Pólya-Szegö inequalities and the explicit example of a domain with sign-changing first eigenfunction. In spite of these fluctuating behaviours, we prove how the Faber-Krahn inequality still holds for any $s>1$ in dimension one.

math.AP↗
arXiv2022

On the shape of the first fractional eigenfunction

We show that the first eigenfunction of the fractional Laplacian ${(-Δ)}^s$, $s\in(1/2,1)$, is superharmonic in the unitary ball up to dimension $11$. To this aim, we also rely on a computer-assisted step to estimate a rather complicated constant depending on the dimension and the power $s$.

math.AP↗
arXiv2021

On the Bernoulli free boundary problems for the half Laplacian and for the spectral half Laplacian

We study the exterior and interior Bernoulli problems for the half Laplacian and the interior Bernoulli problem for the spectral half Laplacian. We concentrate on the existence and geometric properties of solutions. Our main results are the following. For the exterior Bernoulli problem for the half Laplacian, we show that under starshapedness assumptions on the data the free domain is starshaped. For the interior Bernoulli problem for the spectral half Laplacian, we show that under convexity assumptions on the data the free domain is convex and we prove a Brunn-Minkowski inequality for the Bernoulli constant. For Bernoulli problems for the half Laplacian we use a variational approach, whereas for Bernoulli problem for the spectral half Laplacian we use the Beurling method based on subsolutions.

math.AP↗
arXiv2021

Symmetry of odd solutions to equations with fractional Laplacian

We present a symmetry result to solutions of equations involving the fractional Laplacian in a domain with at least two perpendicular symmetries. We show that if the solution is continuous, bounded, and odd in one direction such that it has a fixed sign on one side, then it will be symmetric in the perpendicular direction. Moreover, the solution will be monotonic in the part where it is of fixed sign. In addition, we present also a class of examples in which our result can be applied.

math.AP↗
arXiv2021

Small order asymptotics of the Dirichlet eigenvalue problem for the fractional Laplacian

In this article, we study the asymptotics of Dirichlet eigenvalues and eigenfunctions of the fractional Laplacian $(-Δ)^s$ in bounded open Lipschitz sets in the small order limit $s \to 0^+$. While it is easy to see that all eigenvalues converge to $1$ as $s \to 0^+$, we show that the first order correction in these asymptotics is given by the eigenvalues of the logarithmic Laplacian operator, i.e., the singular integral operator with symbol $2\log|ξ|$. By this we generalize a result of Chen and the third author which was restricted to the principal eigenvalue. Moreover, we show that $L^2$-normalized Dirichlet eigenfunctions of $(-Δ)^s$ corresponding to the $k$-th eigenvalue are uniformly bounded and converge to the set of $L^2$-normalized eigenvalues of the logarithmic Laplacian. In order to derive these spectral asymptotics, we need to establish new uniform regularity and boundary decay estimates for Dirichlet eigenfunctions for the fractional Laplacian. As a byproduct, we also obtain corresponding regularity properties of eigenfunctions of the logarithmic Laplacian.

math.AP↗
arXiv2020

A new look at the fractional Poisson problem via the Logarithmic Laplacian

We analyze the $s$-dependence of solutions $u_s$ to the family of fractional Poisson problems $(-Δ)^s u =f$ in $Ω$, $u \equiv 0$ on $\mathbb{R}^N\setminus Ω$ in an open bounded set $Ω\subset \mathbb{R}^N$, $s \in (0,1)$. In the case where $Ω$ is of class $C^2$ and $f \in C^α(\barΩ)$ for some $α>0$, we show that the map $(0,1) \to L^\infty(Ω)$, $s\mapsto u_s$ is of class $C^1$, and we characterize the derivative $\partial_s u_s$ in terms of the logarithmic Laplacian of $f$. As a corollary, we derive pointwise monotonicity properties of the solution map $s \mapsto u_s$ under suitable assumptions on $f$ and $Ω$. Moreover, we derive explicit bounds for the corresponding Green operator on arbitrary bounded domains which are new even for the case $s=1$, i.e., for the local Dirichlet problem $-Δu = f$ in $Ω$, $u \equiv 0$ on $\partial Ω$.

math.AP↗
arXiv2020

Fractional Laplacians on ellipsoids

We show explicit formulas for the evaluation of (possibly higher-order) fractional Laplacians of some functions supported on ellipsoids. In particular, we derive the explicit expression of the torsion function and give examples of $s$-harmonic functions. As an application, we infer that the weak maximum principle fails in eccentric ellipsoids for $s\in(1,\sqrt{3}+3/2)$ in any dimension $n\geq 2$. We build a counterexample in terms of the torsion function times a polynomial of degree 2. Using point inversion transformations, it follows that a variety of bounded and unbounded domains do not satisfy positivity preserving properties and we give some examples.

math.AP↗