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Register a new userDiscretizations of the spectral fractional Laplacian on general domains with Dirichlet, Neumann, and Robin boundary conditions
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In this work, we propose novel discretizations of the spectral fractional Laplacian on bounded domains based on the integral formulation of the operator via the heat-semigroup formalism. Specifically, we combine suitable quadrature formulas of the integral with a finite element method for the approximation of the solution of the corresponding heat equation. We derive two families of discretizations with order of convergence depending on the regularity of the domain and the function on which the spectral fractional Laplacian is acting. Our method does not require the computation of the eigenpairs of the Laplacian on the considered domain, can be implemented on possibly irregular bounded domains, and can naturally handle different types of boundary constraints. Various numerical simulations are provided to illustrate performance of the proposed method and support our theoretical results.
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Let $Omegasubsetmathbb{R}^ u$, $ uge 2$, be a $C^{1,1}$ domain whose boundary $partialOmega$ is either compact or behaves suitably at infinity. For $pin(1,infty)$ and $alpha>0$, define [ Lambda(Omega,p,alpha):=inf_{substack{uin W^{1,p}(Omega) u otequiv 0}}dfrac{displaystyle int_Omega | abla u|^p mathrm{d} x - alphadisplaystyleint_{partialOmega} |u|^pmathrm{d}sigma}{displaystyleint_Omega |u|^pmathrm{d} x}, ] where $mathrm{d}sigma$ is the surface measure on $partialOmega$. We show the asymptotics [ Lambda(Omega,p,alpha)=-(p-1)alpha^{frac{p}{p-1}} - ( u-1)H_mathrm{max}, alpha + o(alpha), quad alphato+infty, ] where $H_mathrm{max}$ is the maximum mean curvature of $partialOmega$. The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.
Consider a bounded domain with the Dirichlet condition on a part of the boundary and the Neumann condition on its complement. Does the spectrum of the Laplacian determine uniquely which condition is imposed on which part? We present some results, conjectures and problems related to this variation on the isospectral theme.
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems associated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.
We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all centrally symmetric domains.
In this paper we deal with spectral optimization for the Robin Laplacian on a family of planar domains admitting parallel coordinates, namely a fixed-width strip built over a smooth closed curve and the exterior of a convex set with a smooth boundary. We show that if the curve length is kept fixed, the first eigenvalue referring to the fixed-width strip is for any value of the Robin parameter maximized by a circular annulus. Furthermore, we prove that the second eigenvalue in the exterior of a convex domain $Omega$ corresponding to a negative Robin parameter does not exceed the analogous quantity for a disk whose boundary has a curvature larger than or equal to the maximum of that for $partialOmega$.
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