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Weth Tobias - One of the best experts on this subject based on the ideXlab platform.
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Small order asymptotics of the Dirichlet Eigenvalue problem for the fractional Laplacian
2021Co-Authors: Feulefack, Pierre Aime, Jarohs Sven, Weth TobiasAbstract:In this article, we study the asymptotics of Dirichlet Eigenvalues and eigenfunctions of the fractional Laplacian $(-\Delta)^s$ 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 $2\log|\xi|$. 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 $(-\Delta)^s$ 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
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Morse index versus radial symmetry for fractional Dirichlet problems
2021Co-Authors: Fall, Mouhamed Moustapha, Feulefack, Pierre Aime, Temgoua, Remi Yvant, Weth TobiasAbstract:In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-\Delta)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet Eigenvalue problem for the operator $(-\Delta)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet Eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Ba\~{n}uelos and Kulczycki.Comment: 18 page
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Morse index versus radial symmetry for fractional Dirichlet problems
2020Co-Authors: Fall, Mouhamed Moustapha, Feulefack, Pierre Aime, Temgoua, Remi Yvant, Weth TobiasAbstract:In this work, we provide an estimate of the Morse index of radially symmetric sign changing bounded weak solutions $u$ to the semilinear fractional Dirichlet problem $$ (-\Delta)^su = f(u)\qquad \text{ in $\mathcal{B}$},\qquad \qquad u = 0\qquad \text{in $\quad\mathbb{R}^{N}\setminus \mathcal{B}$,} $$ where $s\in(0,1)$, $\mathcal{B}\subset \mathbb{R}^N$ is the unit ball centred at zero and the nonlinearity $f$ is of class $C^1$. We prove that for $s\in(1/2,1)$ any radially symmetric sign changing solution of the above problem has a Morse index greater than or equal to $N+1$. If $s\in (0,1/2],$ the same conclusion holds under additional assumption on $f$. In particular, our results apply to the Dirichlet Eigenvalue problem for the operator $(-\Delta)^s$ in $\mathcal{B}$ for all $s\in (0,1)$, and it implies that eigenfunctions corresponding to the second Dirichlet Eigenvalue in $\mathcal{B}$ are antisymmetric. This resolves a conjecture of Kulczycki.Comment: 18 page
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Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions
2020Co-Authors: Bonheure Denis, Santos, Ederson Moreira Dos, Parini Enea, Tavares Hugo, Weth TobiasAbstract:We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0\lambda_2(\Omega)$ (the second Dirichlet Eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $\Omega$ is either a ball or a square. In particular we give a complete description of the solution set for $\lambda\sim \lambda_2(\Omega)$, computing the Morse index of the solutions.Comment: 26 page
Eder Marinho Martins - One of the best experts on this subject based on the ideXlab platform.
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computing the sinp function via the inverse power method
Computational methods in applied mathematics, 2011Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:In this paper, we discuss a new iterative method for computing sinp. This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet Eigenvalue problem for the p-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra and is competitive with other methods available in the literature.
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computing the first Eigenvalue of the p laplacian via the inverse power method
Journal of Functional Analysis, 2009Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:Abstract In this paper, we discuss a new method for computing the first Dirichlet Eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in R n if p > 1 and for any bounded domain in the particular case p = 2 . For p > 2 the method is validated numerically for the square.
Rodney Josue Biezuner - One of the best experts on this subject based on the ideXlab platform.
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computing the sinp function via the inverse power method
Computational methods in applied mathematics, 2011Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:In this paper, we discuss a new iterative method for computing sinp. This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet Eigenvalue problem for the p-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra and is competitive with other methods available in the literature.
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computing the first Eigenvalue of the p laplacian via the inverse power method
Journal of Functional Analysis, 2009Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:Abstract In this paper, we discuss a new method for computing the first Dirichlet Eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in R n if p > 1 and for any bounded domain in the particular case p = 2 . For p > 2 the method is validated numerically for the square.
Stanislav Molchanov - One of the best experts on this subject based on the ideXlab platform.
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geometric characterization of intermittency in the parabolic anderson model
Annals of Probability, 2007Co-Authors: Jurgen Gartner, Wolfgang Konig, Stanislav MolchanovAbstract:We consider the parabolic Anderson problem @tu = u + (x)u on R+ Z d with localized initial condition u(0;x) = 0(x) and random i.i.d. potential . Under the assumption that the distribution of (0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterisation of intermittency: with probability one, as t ! 1, the overwhelming contribution to the total mass P x u(t;x) comes from a slowly increasing number of 'islands' which are located far from each other. These 'islands' are local regions of those high exceedances of the eld in a box of side length 2t log 2 t for which the (local) principal Dirichlet Eigenvalue of the random operator + is close to the top of the spectrum in the box. We also prove that the shape of in these regions is non-random and that u(t; ) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.
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geometric characterization of intermittency in the parabolic anderson model
arXiv: Probability, 2005Co-Authors: Jurgen Gartner, Wolfgang Konig, Stanislav MolchanovAbstract:We consider the parabolic Anderson problem $\partial_tu=\Delta u+\xi(x)u$ on $\mathbb{R}_+\times\mathbb{Z}^d$ with localized initial condition $u(0,x)=\delta_0(x)$ and random i.i.d. potential $\xi$. Under the assumption that the distribution of $\xi(0)$ has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as $t\to\infty$, the overwhelming contribution to the total mass $\sum_xu(t,x)$ comes from a slowly increasing number of ``islands'' which are located far from each other. These ``islands'' are local regions of those high exceedances of the field $\xi$ in a box of side length $2t\log^2t$ for which the (local) principal Dirichlet Eigenvalue of the random operator $\Delta+\xi$ is close to the top of the spectrum in the box. We also prove that the shape of $\xi$ in these regions is nonrandom and that $u(t,\cdot)$ is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.
Grey Ercole - One of the best experts on this subject based on the ideXlab platform.
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computing the sinp function via the inverse power method
Computational methods in applied mathematics, 2011Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:In this paper, we discuss a new iterative method for computing sinp. This function was introduced by Lindqvist in connection with the unidimensional nonlinear Dirichlet Eigenvalue problem for the p-Laplacian. The iterative technique was inspired by the inverse power method in finite dimensional linear algebra and is competitive with other methods available in the literature.
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computing the first Eigenvalue of the p laplacian via the inverse power method
Journal of Functional Analysis, 2009Co-Authors: Rodney Josue Biezuner, Grey Ercole, Eder Marinho MartinsAbstract:Abstract In this paper, we discuss a new method for computing the first Dirichlet Eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in R n if p > 1 and for any bounded domain in the particular case p = 2 . For p > 2 the method is validated numerically for the square.