Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

We develop further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. The theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. In this paper we extend the theory for the so-called \emph{restricted} fractional Laplacian defined on a bounded domain $\Omega$ of $\mathbb R^N$ with zero Dirichlet conditions outside of $\Omega$. As an application, we derive an original proof of the corresponding fractional Faber-Krahn inequality. We also provide a more classical variational proof of the inequality.Comment: arXiv admin note: substantial text overlap with arXiv:1303.297

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Fourier methods for fractional-order operators

This is a survey on the use of Fourier transformation methods in the treatment of boundary problems for the fractional Laplacian $(-\Delta)^a$ (0<a<1), and pseudodifferential generalizations P, over a bounded open set $\Omega$ in $R^n$. The presentation starts at an elementary level. Two points are explained in detail: 1) How the factor $d^a$, with $d(x)=dist(x,d\Omega)$, comes into the picture, related to the fact that the precise solution spaces for the homogeneous Dirichlet problem are so-called a-transmission spaces. 2) The natural definition of a local nonhomogeneous Dirichlet condition $\gamma_0(u/d^{a-1})=\varphi$. We also give brief accounts of some further developments: Evolution problems (for $d_t u - r^+Pu = f(x,t)$) and resolvent problems (for $Pu-\lambda u=f$), also with nonzero boundary conditions. Integration by parts, Green's formula.Comment: 20 pages. Prepared for the Proceedings of the RIMS Symposium "Harmonic Analysis and Nonlinear Partial Differential equations", July 11-13, 2022, in the RIMS Kokyuroku Bessatsu series. Small improvements of formulation

An efficient implementation of an implicit FEM scheme for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a modelling tool for processes with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues, which impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractional-in-space reaction diffusion equations using the finite element method on both structured and unstructured grids, and robust techniques for computing the fractional power of a matrix times a vector. Our approach is show-cased by solving the fractional Fisher and fractional Allen-Cahn reaction-diffusion equations in two and three spatial dimensions, and analysing the speed of the travelling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator

DATA-DRIVEN MODELING AND SIMULATIONS OF SEISMIC WAVES

In recent decades, nonlocal models have been proved to be very effective in the study of complex processes and multiscale phenomena arising in many fields, such as quantum mechanics, geophysics, and cardiac electrophysiology. The fractional Laplacian(−Δ)/2 can be reviewed as nonlocal generalization of the classical Laplacian which has been widely used for the description of memory and hereditary properties of various material and process. However, the nonlocality property of fractional Laplacian introduces challenges in mathematical analysis and computation. Compared to the classical Laplacian, existing numerical methods for the fractional Laplacian still remain limited. The objectives of this research are to develop new numerical methods to solve nonlocal models with fractional Laplacian and apply them to study seismic wave modeling in both homogeneous and heterogeneous media. To this end, we have developed two classes of methods: meshfree pseudospectral method and operator factorization methods. Compared to the current state-of-the-art methods, both of them can achieve higher accuracy with less computational complexity. The operator factorization methods provide a general framework, allowing one to achieve better accuracy with high-degree Lagrange basis functions. The meshfree pseudospectral methods based on global radial basis functions can solve both classical and fractional Laplacians in a single scheme which are the first compatible methods for these two distinct operators. Numerical experiments have demonstrated the effectiveness of our methods on various nonlocal problems. Moreover, we present an extensive study of the variable-order Laplacian operator (−Δ)(x)/2 by using meshfree methods both analytically and numerically. Finally, we apply our numerical methods to solve seismic wave modeling and study the nonlocal effects of fractional wave equation --Abstract, p. i

De Giorgi's approach to hyperbolic Cauchy problems: the case of nonhomogeneous equations

In this paper we discuss an extension of some results obtained by E. Serra and P. Tilli, in [Serra&Tilli '12, Serra&Tilli '16], concerning an original conjecture by E. De Giorgi ([De Giorgi '96, De Giorgi '06]) on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to extend the techniques developed by Serra and Tilli for homogeneous hyperbolic nonlinear PDEs to the nonhomogeneous case, thus proving that the idea of De Giorgi yields in fact an effective approach to investigate general hyperbolic equations.Comment: 24 page