19 results on '"Nonlocal operators"'
Search Results
2. Maximum principles and related problems for a class of nonlocal extremal operators
- Author
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Isabeau Birindelli, Giulio Galise, and Delia Schiera
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Fully nonlinear degenerate elliptic PDE ,Maximum and comparison principles ,Nonlocal operators ,Eigenvalue problem ,Mathematics - Analysis of PDEs ,Applied Mathematics ,FOS: Mathematics ,Analysis of PDEs (math.AP) - Abstract
We study the validity of the comparison and maximum principles and their relation with principal eigenvalues, for a class of degenerate nonlinear operators that are extremal among operators with one-dimensional fractional diffusion.
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- 2021
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3. Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order
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Jamil Chaker and Minhyun Kim
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Mathematics - Analysis of PDEs ,Mathematics::Probability ,divergence form ,measures ,Mathematics::Analysis of PDEs ,FOS: Mathematics ,35B65, 47G20, 31B05, 42B25 ,regularity theory ,anisotropic ,weak Harnack inequality ,Analysis ,nonlocal operators ,Analysis of PDEs (math.AP) - Abstract
We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local H\"older estimate., Comment: 28 pages, 2 figures, slight modifications and corrections
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- 2021
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4. Exponential Convergence of $hp$ FEM for Spectral Fractional Diffusion in Polygons
- Author
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Banjai, Lehel, Melenk, Jens M., and Schwab, Christoph
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Applied Mathematics ,n-widths ,26A33, 65N12, 65N30 ,Numerical Analysis (math.NA) ,Geometric corner refinement ,Nonlocal operators ,Anisotropic hp–refinement ,Computational Mathematics ,Exponential convergence ,Fractional diffusion ,Dunford-Taylor calculus ,FOS: Mathematics ,Mathematics - Numerical Analysis - Abstract
For the spectral fractional diffusion operator of order 2s, s ∈ (0,1), in bounded, curvilinear polygonal domains Ω ⊂ R2 we prove exponential convergence of two classes of hp discretizations under the assumption of analytic data (coefficients and source terms, without any boundary compatibility), in the natural fractional Sobolev norm Hs(Ω). The first hp discretization is based on writing the solution as a co-normal derivative of a 2+1-dimensional local, linear elliptic boundary value problem, to which an hp-FE discretization is applied. A diagonalization in the extended variable reduces the numerical approximation of the inverse of the spectral fractional diffusion operator to the numerical approximation of a system of local, decoupled, second order reaction-diffusion equations in Ω. Leveraging results on robust exponential convergence of hp-FEM for second order, linear reaction diffusion boundary value problems in Ω, exponential convergence rates for solutions u ∈ Hs(Ω) of Lsu = f follow. Key ingredient in this hp-FEM are boundary fitted meshes with geometric mesh refinement towards ∂Ω. The second discretization is based on exponentially convergent numerical sinc quadrature approximations of the Balakrishnan integral representation of L−s combined with hp-FE discretizations of a decoupled system of local, linear, singularly perturbed reaction-diffusion equations in Ω. The present analysis for either approach extends to (polygonal subsets M˜ of) analytic, compact 2-manifolds M, parametrized by a global, analytic chart χ with polygonal Euclidean parameter domain Ω ⊂ R2. Numerical experiments for model problems in nonconvex polygonal domains and with incompatible data confirm the theoretical results. Exponentially small bounds on Kolmogorov n-widths of solution sets for spectral fractional diffusion in curvilinear polygons and for analytic source terms are deduced., Numerische Mathematik, 153 (1), ISSN:0029-599X, ISSN:0945-3245
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- 2020
5. A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian
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Juan Pablo Borthagaray, Francisco M. Bersetche, and Gabriel Acosta
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Matemáticas ,Context (language use) ,010103 numerical & computational mathematics ,Space (mathematics) ,01 natural sciences ,Matemática Pura ,purl.org/becyt/ford/1 [https] ,FINITE ELEMENTS ,Simple (abstract algebra) ,FOS: Mathematics ,Code (cryptography) ,Mathematics - Numerical Analysis ,0101 mathematics ,MATLAB ,NONLOCAL OPERATORS ,Mathematics ,computer.programming_language ,Dirichlet problem ,FRACTIONAL LAPLACIAN ,purl.org/becyt/ford/1.1 [https] ,Numerical Analysis (math.NA) ,Finite element method ,010101 applied mathematics ,Algebra ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Variety (universal algebra) ,computer ,CIENCIAS NATURALES Y EXACTAS - Abstract
In \cite{AcostaBorthagaray}, a complete $n$-dimensional finite element analysis of the homogeneous Dirichlet problem associated to a fractional Laplacian was presented. Here we provide a comprehensive and simple 2D {\it MATLAB}\textsuperscript{\textregistered} finite element code for such a problem. The code is accompanied with a basic discussion of the theory relevant in the context. The main program is written in about 80 lines and can be easily modified to deal with other kernels as well as with time dependent problems. The present work fills a gap by providing an input for a large number of mathematicians and scientists interested in numerical approximations of solutions of a large variety of problems involving nonlocal phenomena in two-dimensional space., Source code is available upon request
- Published
- 2017
6. Space-Time fractional diffusion in cell movement models with delay
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Kevin J. Painter, Heiko Gimperlein, Gissell Estrada-Rodriguez, Jakub Stocek, Universitat Politècnica de Catalunya. Departament de Matemàtiques, and Universitat Politècnica de Catalunya. TP-EDP - Grup de Teoria de Funcions i Equacions en Derivades Parcials
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FOS: Physical sciences ,Brain tissue ,Lévy process ,nonlocal operators ,immune cells ,velocity-jump model ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,Fractional diffusion ,Physics - Biological Physics ,Mathematics - Numerical Analysis ,Physics ,Física [Àrees temàtiques de la UPC] ,Macroscopic model ,Movement (music) ,Applied Mathematics ,Space time ,Mathematical analysis ,Immune cells ,Matemàtiques i estadística [Àrees temàtiques de la UPC] ,Física ,Numerical Analysis (math.NA) ,Nonlocal operators ,Lévy flight ,Biological Physics (physics.bio-ph) ,Modeling and Simulation ,Velocity jump model ,Matemàtica ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
The movement of organisms and cells can be governed by occasional long distance runs, according to an approximate L\'evy walk. For T cells migrating through chronically-infected brain tissue, runs are further interrupted by long pauses, and the aim here is to clarify the form of continuous model equations which describe such movements. Starting from a microscopic velocity-jump model based on experimental observations, we include power-law distributions of run and waiting times and investigate the relevant parabolic limit from a kinetic equation for resting and moving individuals. In biologically relevant regimes we derive nonlocal diffusion equations, including fractional Laplacians in space and fractional time derivatives. Its analysis and numerical experiments shed light on how the searching strategy, and the impact from chemokinesis responses to chemokines, shorten the average time taken to find rare targets in the absence of direct guidance information such as chemotaxis., Comment: 25 pages, 8 figures, Mathematical Models and Methods in Applied Sciences (2019)
- Published
- 2019
7. Maximum principles for nonlocal parabolic Waldenfels operators
- Author
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Jinqiao Duan, Qiao Huang, and Jiang-Lun Wu
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Class (set theory) ,General Mathematics ,fokker–planck equations ,Markov process ,Context (language use) ,Type (model theory) ,01 natural sciences ,Lévy process ,waldenfels operators ,nonlocal operators ,Stochastic differential equation ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mathematics::Probability ,0103 physical sciences ,FOS: Mathematics ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics ,weak and strong maximum principles ,integro-partial differential equations ,Markov chain ,lcsh:Mathematics ,Probability (math.PR) ,010102 general mathematics ,Mathematical analysis ,lcsh:QA1-939 ,35B50(Primary), 35R09 (Secondary) ,symbols ,stochastic differential equations with α-stable lévy processes ,010307 mathematical physics ,Mathematics - Probability ,Analysis of PDEs (math.AP) - Abstract
As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `parabolic' equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with $\alpha$-stable L\'evy processes are presented to illustrate the maximum principles., Comment: 38 pages, 3 figures
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- 2019
8. A note on homogeneous Sobolev spaces of fractional order
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Ariel Martin Salort and Lorenzo Brasco
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Pure mathematics ,Matemáticas ,Open set ,Mathematics::Analysis of PDEs ,Poincaré inequality ,Fractional Sobolev spaces, Nonlocal operators, Poincaré inequality, Real interpolation ,01 natural sciences ,NO ,Matemática Pura ,symbols.namesake ,Mathematics - Analysis of PDEs ,Real interpolation ,0103 physical sciences ,FRACTIONAL SOBOLEV SPACES ,FOS: Mathematics ,0101 mathematics ,NONLOCAL OPERATORS ,POINCARÉ INEQUALITY ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Fractional Sobolev spaces ,Nonlocal operators ,Functional Analysis (math.FA) ,Sobolev space ,Mathematics - Functional Analysis ,REAL INTERPOLATION ,Homogeneous ,Norm (mathematics) ,symbols ,Interpolation space ,010307 mathematical physics ,CIENCIAS NATURALES Y EXACTAS ,Interpolation theory ,Analysis of PDEs (math.AP) - Abstract
We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\u{\i} norm. We compare it to the fractional Sobolev space obtained by the $K-$method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible., Comment: 33 pages, 2 figures
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- 2018
9. On fractional Hardy inequalities in convex sets
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Eleonora Cinti, Lorenzo Brasco, Brasco, Lorenzo, Cinti, Eleonora, Dipartimento di Matematica e Informatica [Ferrara], Università degli Studi di Ferrara (UniFE), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Dipartimento di Matematica [Bologna], Alma Mater Studiorum Università di Bologna [Bologna] (UNIBO), MINECO MTM2014-52402-C3-1-P, Catalan research group 2014 SGR 1083, European Project: 339958,EC:FP7:ERC,ERC-2013-ADG,COMPAT(2014), Dipartimento di Matematica e Informatica = Department of Mathematics and Computer Science [Ferrara] (DMCS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), and Università degli Studi di Ferrara = University of Ferrara (UniFE)
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Inequality ,media_common.quotation_subject ,Convex set ,Mathematics::Analysis of PDEs ,Boundary (topology) ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,NO ,Combinatorics ,Mathematics - Analysis of PDEs ,Hardy inequality ,39B72 ,35R11, 46E35 ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Nonlocal operator ,0101 mathematics ,[MATH]Mathematics [math] ,Discrete Mathematics and Combinatoric ,media_common ,Mathematics ,Subharmonic function ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,Fractional Sobolev spaces ,Order (ring theory) ,Analysi ,Function (mathematics) ,Nonlocal operators ,Fractional Sobolev space ,Functional Analysis (math.FA) ,010101 applied mathematics ,Mathematics - Functional Analysis ,Constant (mathematics) ,Analysis ,Analysis of PDEs (math.AP) - Abstract
We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodecki\u{\i} spaces of order $(s,p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1, Comment: 25 pages, 3 figures
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- 2018
10. Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
- Author
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Espen R. Jakobsen, Jørgen Endal, and Félix del Teso
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a priori estimates ,fast diffusion equation ,Type (model theory) ,01 natural sciences ,nonlocal operators ,Mathematics - Analysis of PDEs ,porous medium equation ,Convergence (routing) ,FOS: Mathematics ,Uniqueness ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics ,nonlinear degenerate diffusion ,Numerical Analysis ,convergence ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Mathematical analysis ,Stefan problem ,existence ,uniqueness ,Numerical Analysis (math.NA) ,010101 applied mathematics ,Computational Mathematics ,Nonlinear system ,Fully discrete numerical schemes ,fractional Laplacian ,Laplacian ,Porous medium ,Laplace operator ,distributional solutions ,Analysis of PDEs (math.AP) - Abstract
\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric L\'evy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=\Delta$ or $\mathfrak{L}=-(-\Delta)^{\frac\alpha2}$, $\alpha\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $\varphi$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{loc}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems., Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway - ERCIM “Alain Bensoussan” Fellowship programme - “Juan de la Cierva - formación” program (FJCI-2016-30148)
- Published
- 2018
11. Tensor FEM for spectral fractional diffusion
- Author
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Abner J. Salgado, Christoph Schwab, Ricardo H. Nochetto, Jens Markus Melenk, Lehel Banjai, and Enrique Otárola
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Pure mathematics ,Applied Mathematics ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Bochner space ,01 natural sciences ,Fractional diffusion ,Nonlocal operators ,Weighted Sobolev spaces ,Regularity estimates ,Finite elements ,Anisotropic hp-refinement ,Corner refinement ,Sparse grids ,Exponential convergence ,Sobolev space ,Computational Mathematics ,Elliptic operator ,symbols.namesake ,Tensor product ,Computational Theory and Mathematics ,Norm (mathematics) ,Dirichlet boundary condition ,Bounded function ,FOS: Mathematics ,symbols ,Mathematics - Numerical Analysis ,0101 mathematics ,Analysis ,Analytic function ,Mathematics - Abstract
We design and analyze several finite element methods (FEMs) applied to the Caffarelli–Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^d$$ with $$d=1,2$$ . For the solution to the Caffarelli–Silvestre extension, we establish analytic regularity with respect to the extended variable $$y\in (0,\infty )$$ . Specifically, the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to y, taking values in corner-weighted Kondrat’ev-type Sobolev spaces in $$\varOmega $$ . In $$\varOmega \subset {\mathbb {R}}^2$$ , we discretize with continuous, piecewise linear, Lagrangian FEM ( $$P_1$$ -FEM) with mesh refinement near corners and prove that the first-order convergence rate is attained for compatible data $$f\in \mathbb {H}^{1-s}(\varOmega )$$ with $$0
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- 2017
12. Nonlocal problems with Neumann boundary conditions
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Enrico Valdinoci, Xavier Ros-Oton, and Serena Dipierro
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General Mathematics ,010102 general mathematics ,Boundary (topology) ,Neumann problem ,Mathematics::Spectral Theory ,Nonlocal operators ,01 natural sciences ,Parabolic partial differential equation ,Domain (mathematical analysis) ,010101 applied mathematics ,Fractional Laplacian ,35R11 ,Mathematics - Analysis of PDEs ,Simple (abstract algebra) ,Neumann boundary condition ,FOS: Mathematics ,Applied mathematics ,Heat equation ,Limit (mathematics) ,60G22 ,0101 mathematics ,Constant (mathematics) ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We introduce a new Neumann problem for the fractional Laplacian arising from a simple probabilistic consideration, and we discuss the basic properties of this model. We can consider both elliptic and parabolic equations in any domain. In addition, we formulate problems with nonhomogeneous Neumann conditions, and also with mixed Dirichlet and Neumann conditions, all of them having a clear probabilistic interpretation. We prove that solutions to the fractional heat equation with homogeneous Neumann conditions have the following natural properties: conservation of mass inside Ω, decreasing energy, and convergence to a constant as t→∞. Moreover, for the elliptic case we give the variational formulation of the problem, and establish existence of solutions. We also study the limit properties and the boundary behavior induced by this nonlocal Neumann condition. For concreteness, one may think that our nonlocal analogue of the classical Neumann condition ∂νu=0 on~∂Ω consists in the nonlocal prescription ∫Ωu(x)−u(y)|x−y|n+2sdy=0for x∈Rn∖Ω. We made an effort to keep all the arguments at the simplest possible technical level, in order to clarify the connections between the different scientific fields that are naturally involved in the problem, and make the paper accessible also to a wide, non-specialistic public (for this scope, we also tried to use and compare different concepts and notations in a somehow more unified way).
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- 2017
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13. Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type
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Espen R. Jakobsen, Jørgen Endal, and Félix del Teso
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Anomalous diffusion ,General Mathematics ,01 natural sciences ,nonlocal operators ,Mathematics - Analysis of PDEs ,porous medium equation ,FOS: Mathematics ,Initial value problem ,Mathematics - Numerical Analysis ,Uniqueness ,0101 mathematics ,Mathematics ,nonlinear degenerate diffusion ,convergence ,Operator (physics) ,010102 general mathematics ,Degenerate energy levels ,Mathematical analysis ,existence ,Stefan problem ,uniqueness ,Numerical Analysis (math.NA) ,stability ,010101 applied mathematics ,Elliptic operator ,Bounded function ,continuous dependence ,local limits ,fractional Laplacian ,distributional solutions ,numerical approximation ,Analysis of PDEs (math.AP) - Abstract
We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^\mu [\varphi (u)]=0$. Here $\mathcal{L}^\mu$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $\varphi:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations., Comment: To appear in "Advances in Mathematics"
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- 2017
14. Regularity of solutions to anisotropic nonlocal equations
- Author
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Jamil Chaker
- Subjects
Harmonic functions ,General Mathematics ,010102 general mathematics ,Probability (math.PR) ,Order (ring theory) ,Condensed Matter::Mesoscopic Systems and Quantum Hall Effect ,01 natural sciences ,010104 statistics & probability ,Stochastic differential equation ,Support theorem ,Harmonic function ,Mathematics::Probability ,Bounded function ,FOS: Mathematics ,60J75, 60H10, 31B05, 60G52 ,Anisotropy ,Nonlocal Operators ,0101 mathematics ,Jump processes ,Mathematics - Probability ,Mathematics ,Mathematical physics ,Holder continuity - Abstract
We study harmonic functions associated to systems of stochastic differential equations of the form $dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots+A_{id}(X_{t-})dZ_t^d$, $i\in\{1,\dots,d\}$, where $Z_t^j$ are independent one-dimensional symmetric stable processes with indices $\alpha_j\in(0,2)$, $j\in\{1,\dots,d\}$. In this article we prove H\"older regularity of bounded harmonic functions with respect to solutions to such systems., Comment: 18 pages
- Published
- 2016
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15. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential
- Author
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Jürgen Sprekels, Pierluigi Colli, and Gianni Gilardi
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State variable ,Control and Optimization ,Parameter identification ,phase field systems ,nonlinear inverse problem ,Subderivative ,01 natural sciences ,nonlocal operators ,symbols.namesake ,Mathematics - Analysis of PDEs ,Indicator function ,Differential inclusion ,semilinear wave equation ,FOS: Mathematics ,Applied mathematics ,0101 mathematics ,Mathematics - Optimization and Control ,49K20 ,Mathematics ,first-order necessary optimality conditions ,Series (mathematics) ,Tikhonov regularization ,Applied Mathematics ,010102 general mathematics ,approximate source condition ,Optimal control ,Distributed optimal control ,conditional stability ,010101 applied mathematics ,Nonlinear system ,74A15 ,Optimization and Control (math.OC) ,Modeling and Simulation ,Lagrange multiplier ,35K55, 49K20, 74A15 ,35K55 ,symbols ,double obstacle potentials ,Analysis of PDEs (math.AP) - Abstract
This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors; the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called `deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential., Comment: Key words: distributed optimal control, phase field systems, double obstacle potentials, nonlocal operators, first-order necessary optimality conditions. The interested reader can also see the related preprints arXiv:1511.04361 and arXiv:1605.07801 whose results are recalled and used for the analysis carried out in this paper
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- 2016
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16. Distributed optimal control of a nonstandard nonlocal phase field system
- Author
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Jürgen Sprekels, Pierluigi Colli, and Gianni Gilardi
- Subjects
Field (physics) ,Distributed optimal control| nonlinear phase field systems| nonlocal operators| first-ordernecessary optimality conditions ,lcsh:Mathematics ,General Mathematics ,Phase (waves) ,Type (model theory) ,Nonlinear phase field systems ,Nonlocal operators ,lcsh:QA1-939 ,Optimal control ,Distributed optimal control ,First-order necessary optimality conditions ,Mathematics - Analysis of PDEs ,74A15 ,Optimization and Control (math.OC) ,35K55, 49K20, 74A15 ,35K55 ,FOS: Mathematics ,Applied mathematics ,Atomic lattice ,Mathematics - Optimization and Control ,49K20 ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
We investigate a distributed optimal control problem for a nonlocal phase field model of viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion that has been studied in a series of papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinear parabolic equation coupled to an ordinary differential equation. The latter equation contains both nonlocal and singular terms that render the analysis difficult. Standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality., Comment: 38 Pages. Key words: distributed optimal control, nonlinear phase field systems, nonlocal operators, first-order necessary optimality conditions
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- 2016
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17. Local density of Caputo-stationary functions in the space of smooth functions
- Author
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Claudia Bucur and Bucur, C
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Control and Optimization ,Derivative ,010502 geochemistry & geophysics ,Space (mathematics) ,01 natural sciences ,Caputo stationary ,Mathematics - Analysis of PDEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Nonlocal operator ,0101 mathematics ,Fractional derivative ,Nonlocal operators ,Initial point ,0105 earth and related environmental sciences ,Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,Function (mathematics) ,Fractional calculus ,Computational Mathematics ,Control and Systems Engineering ,Mathematics - Classical Analysis and ODEs ,Analysis of PDEs (math.AP) - Abstract
We consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any $C^k\big([0,1]\big)$ function can be approximated in $[0,1]$ by a a function that is Caputo-stationary in $[0,1]$, with initial point $a, 19 pages, 4 figures
- Published
- 2015
18. Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients
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Fabio Punzo and Enrico Valdinoci
- Subjects
Sub- supersolutions ,35B99 ,Control and Optimization ,media_common.quotation_subject ,35B51 ,Evolution equations ,Space (mathematics) ,01 natural sciences ,Mathematics - Analysis of PDEs ,35K67 ,FOS: Mathematics ,Uniqueness ,0101 mathematics ,Nonlocal operators ,Control and Systems Engineering ,Computational Mathematics ,Mathematics ,media_common ,Existence and uniqueness results ,010102 general mathematics ,Mathematical analysis ,35B40 ,fractional parabolic and elliptic equations ,Computational mathematics ,Infinity ,Parabolic partial differential equation ,35K61 ,010101 applied mathematics ,Analysis of PDEs (math.AP) - Abstract
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed point-wise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behavior of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.
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- 2015
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19. Weyl-type laws for fractional p-eigenvalue problems
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Marco Squassina and Antonio Iannizzotto
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Pure mathematics ,Weyl type law ,General Mathematics ,Mathematical analysis ,Type (model theory) ,Mathematics::Spectral Theory ,Weyl estimates ,Domain (mathematical analysis) ,Settore MAT/05 - ANALISI MATEMATICA ,nonlocal operators ,Mathematics - Analysis of PDEs ,Bounded function ,FOS: Mathematics ,35P15, 35P30, 35R11 ,Eigenvalues and eigenvectors ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
We prove an asymptotic estimate for the growth of variational eigenvalues of fractional p-Laplacian eigenvalue problems on a smooth bounded domain., Comment: 10 pages
- Published
- 2013
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