Abstract : The focus of the minisymposium will be on different aspects of nonlocal operators including regularity and numerical analysis of solutions to equations driven by fractional and nonlocal operators. In the recent years nonlocal models showed effectivity in describing phenomena involving different singularities. We aims to bring together leading experts and young researchers interested in nonlocality, in particular for nonlinear problems, including:
Numerics and Scientific Computing, Modeling and Applications, Analysis of Partial Differential Equations, Calculus of Variations.
Abstract : The essential feature of many models with non-standard growth is the possible presence of Lavrentiev gap and related lack of regularity, non-density of smooth functions in the corresponding energy space. Finding assumptions for the presence of Lavrentiev phenomena is in particular important for regularity theory. We show that nonlocal and local-nonlocal models enjoy the presence of energy gap. We obtain the optimal conditions separating the regular case from the one with Lavrentiev gap for the different types of nonlocal and mixed local-nonlocal double phase models. The obtained conditions show the sharpness of resent regularity results for nonlocal double-phase problems.
[04387] Kacanov Iteration
Format : Talk at Waseda University
Author(s) :
Lars Diening (Bielefeld University)
Anna Kh. Balci (Bielefeld University)
Johannes Storn (Bielefeld University)
Abstract : The p-Laplace equation is one of the model equations for non-linear
problems. Due to its non-linearity it is quite challenging to
approximate its solution numerically in particular in the
degenerate/singular case. Standard methods like gradient descent or
Newton's method have significant problems to approximate the
solution. We present an iterative, linear method that allows to solve
the p-Laplace equation efficiently both for small and large exponents.
[04258] Regularity results for fractional nonlocal equation with nonstandard growth and differentiability
Format : Talk at Waseda University
Author(s) :
Jihoon Ok (Sogang University)
Abstract : We discuss on nonlocal problems with nonstandard growth and differentiability. In particular, we introduce local boundedness and Hölder continuity for nonlocal double phase problems and nonlocal problems with variable growth and differentiability, by identifying sharp assumptions on parameters and functions characterizing these nonlocal problems.
[05242] BBM-type theorem for fractional Sobolev spaces with variable exponents
Format : Talk at Waseda University
Author(s) :
Minhyun Kim (Hanyang University)
Abstract : A Bourgain–Brezis–Mironescu-type theorem for fractional Sobolev spaces with variable exponents is established for sufficiently regular functions. We prove, however, that a limiting embedding theorem for these spaces fails to hold in general.
[05189] Time fractional gradient flows: Theory and numerics
Author(s) :
Abner J Salgado (University of Tennessee)
Wenbo Li (Chinese Academy of Sciences)
Abstract : We develop the theory of fractional gradient flows: an evolution aimed at the minimization of a convex, lower semicontinuous energy, with memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce the notion of energy solutions, for which we provide existence, uniqueness and certain regularizing effects. We also consider Lipschitz perturbations of this energy. For these problems we provide an a posteriori error estimate and show its reliability. This estimate depends only on the problem data, and imposes no constraints between consecutive time-steps. On the basis of this estimate we provide an a priori error analysis that makes no assumptions on the smoothness of the solution.
[05258] Semiconvexity estimates for integro-differential equations
Format : Talk at Waseda University
Author(s) :
Marvin Weidner (Universitat de Barcelona)
Abstract : The Bernstein technique is a classical tool to establish derivative estimates for solutions to a large class of elliptic and parabolic equations. It is based on the maximum principle applied to suitable auxiliary functions.
We explain how the Bernstein technique can be extended to integro-differential equations. As an application, we establish semiconvexity estimates for solutions to the nonlocal obstacle problem, the optimal regularity of the solution and
the regularity of the free boundary. Based on a joint work with Xavier Ros-Oton and Damià Torres-Latorre.
[05644] Gradient flow for symmetric nonlocal Lévy operators
Format : Talk at Waseda University
Author(s) :
Guy Foghem (Technische Universität Dresden )
Markus Schmidtchen (Technische Universität Dresden)
David Padilla-Garza (Technische Universität Dresden)
Abstract : We study a nonlocal continuity equation for the probability density with pressure driven by a symmetric nonlocal Lévy operator. The class of nonlocal operators under consideration appear as a generalization of the classical fractional Laplace operator. We construct weak solutions with respect to the associated homogeneous nonlocal Sobolev space using the minimizing movement scheme technique. The lack of interpolation techniques brings up some difficulties and renders our approach different from the usual interpolation techniques.
[04692] The Spatially Variant Spectral Fractional Laplacian: Analytical Aspects and Parameter Selection
Format : Online Talk on Zoom
Author(s) :
Carlos Rautenberg (George Mason University)
Andrea Ceretani (University of Buenos Aires)
Abstract : We consider a variational definition for the spatially variant (spectral) fractional Laplacian and study the well-posedness of the associated Poisson’s equation. The state space for the elliptic problem relies on non-standard Sobolev spaces with weights that are not of Muckenhoupt-type. The increased regularity of solutions is established together with the effectiveness of the fractional operator as a regularization for inverse problems. The latter leads to the optimal selection of the fractional order in image reconstruction.
