The seminar takes place on selected** Wednesdays 8-9 am German time**.

Zoom: https://uni-bielefeld.zoom.us/j/96823879152?pwd=VmRmSUtQMUpYdE44U1UwdmxwSTB5dz09

ID: 968 2387 9152

Kenncode: 2235

**Jörn Kommer (Bielefeld)**: A tensor product approach to non-local differential complexes

Abstract: We start the talk by giving a brief introduction to classical differential forms and the de Rham complex on a manifold.

We then define differential complexes of Alexander-Spanier(-Kolmogoroff) type on metric measure spaces associated with unbounded non-local operators like the fractional Laplacian. These complexes are Hilbert complexes, and they can be seen as non-local counterparts of the de Rham complex. We can obtain self-adjoint non-local analogues of Hodge Laplacians. If time permits, we prove a Mayer-Vietoris principle and a Poincaré lemma, and verify that in the compact Riemannian manifold case the de Rham cohomology can be recovered.

This talk is based on joint work with Michael Hinz.

**Kyeong Song (Seoul)**: Regularity results for double phase problems involving nonlocal operators

Abstract: In this talk, we investigate the De Giorgi-Nash-Moser theory for nonlocal problems and mixed local and nonlocal problems. We first recall known regularity results for fractional p-Laplacian problems in the context of the calculus of variations. We then introduce two kinds of double phase problems; one is a nonlocal double phase problem, and the other is a mixed local and nonlocal double phase problem. We present our recent results about Hölder regularity and Harnack inequality for such problems under sharp assumptions.

**Myeongju Kang (Seoul)**: t.b.a.

**Patricia Alonso Ruiz (Texas)**: Sobolev embeddings in Dirichlet spaces

Abstract: A classical Sobolev embedding in $\mathbb{R}^n$ asserts that functions in the Sobolev space $W^{1,p}(\mathbb{R}^n)$ with $1 \leq p < n$ belong to a suitable $L^q$-space with an explicit optimal exponent q that depends on n and p. How does this embedding read in the context of Dirichlet spaces? This talk will provide an answer in the context of metric measure spaces equipped with a regular Dirichlet form. In particular, we will discover how the optimal exponent depends on the Hausdorff dimension, the walk dimension, and a further invariant of the space. The main assumptions on the underlying space are heat kernel estimates and a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in classical settings as well as in fractals like (infinite) Sierpinski gaskets and carpets. The talk is based on joint work with F. Baudoin.

**Ho-Sik Lee (Seoul)**: Global Maximal Regularity for Equations with Degenerate Weights.

Abstract: Elliptic equations are usually studied under the uniform ellipticity assumption. In this talk, I allow the degeneracy to the uniform ellipticity and consider the elliptic equations with degenerate weights, where the weight is given by a function belonging to a certain Muckenhoupt class. Then I introduce the result of global Calderon-Zygmund type estimates, which are obtained by imposing a small BMO condition on the logarithm of the degenerate weight and a small Lipschitz constant condition on the boundary of the domain. I also discuss the sharpness of our result by an example.

**Patrícia Gonçalves (Lisbon)**: From particle systems to partial differential equations.

Abstract: In this seminar I will describe the derivation of partial differential equations that rule the space-time evolution of the conserved quantity (ies) of stochastic processes. The random dynamics conserves a quantity (as the total mass) that has a non-trivial evolution in space and time. The goal is to describe the connection between the macroscopic (continuous) equations and the microscopic (discrete) system of random particles. The former can be either PDEs or stochastic PDEs depending on whether one is looking at the law of large numbers or the central limit theorem scaling; while the latter is a collection of particles that move randomly according to a transition probability and rings of Poisson processes. I will focus on a particular particle system, namely, the exclusion process and explain that depending on the choice of the jump rates we get to different partial differential equations.

**Jaehun Lee (Seoul)**: General law of iterated logarithm for Markov processes

Abstract : Law of iterated logarithm for a stochastic process describes the magnitude of the fluctuations of its sample path behaviors. In this talk, we discuss the law of iterated logarithm for continuous-time Markov processes, which cover a large class of subordinated diffusions, jump processes with mixed polynomial local growth, jump processes with singular jumping kernels and random conductance models with long range jumps. We establish the law of iterated logarithms for the distance and occupation time, in terms of two functions concerned with the first exit time from balls, and uniform bounds on the tails of jumping measure. Especially, we concentrate on the sharp sufficient conditions for both small and large time. This is joint work with Soobin Cho and Panki Kim.

This is the joint work with Soobin Cho and Panki Kim.

**Tabea Tscherpel (Bielefeld)**: On the Sobolev stability of the $L^2$-Projection mapping to finite element spaces

Abstract: The $L^2$-projection mapping to Lagrange finite element spaces is a crucial tool in numerical analysis. Especially when dealing with evolution problems Sobolev stability of the $L^2$-projection is needed. While the proof of Sobolev stability is straightforward for uniform meshes, it is much more challenging for graded meshes. We tackle this difficulty by investigating the decay properties of the $L^2$-projection analytically. Inspired by the miracle of extrapolation we introduce a maximal operator. This allows us to obtain equivalence of decay and certain weighted estimates, that eventually lead to Sobolev stability. We present recent results in 2D and 3D for a wide range of polynomial degrees and graded meshes. Those include a large class of meshes used in practical applications.

This is joint work with Lars Diening and Johannes Storn from Bielefeld University (Germany).

**Minhyun Kim (Bielefeld)**: Local regularity theory for nonlocal equations

Abstract: We introduce several nonlocal equations which correspond to classical local models, attempting to develop parallel or unified theory. We study local regularity properties such as local boundedness, weak Harnack inequality, and local Hölder regularity of weak solutions to these nonlocal equations.