Titles and Abstracts

Nonlocal Lavrentiev Phenomenon

Anna Balci (Charles University, Prague, Czechia)
Abstract: We present a general framework for constructing examples of Lavrentiev energy gap in nonlocal problems. This framework is applied to various nonlocal and mixed models of the double-phase type. Additionally, we explore the density of smooth functions.

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Optimal control approach to mean field system

Viorel Barbu (Alexandru Iona Cuza University of Iași, Romanian Academy in Bucharest, Romania)
Abstract: The solution to forward-backward system which describes the mean field game model is obtained as solution to an optimal control problem governed by a linear Fokker-Planck equation with control in drift term.

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Random attractors for boundary value problems with dynamical boundary conditions (Poster)

Simon Bau (University of Konstanz, Germany)
Abstract: We study regularity aspects of similinear evolution equations with dynamical boundary conditions and the dynamics they generate under the influence of noise. Before we consider the stochastic setting, it is essential to understand the determinisitic case first. Here, we define an appropriate \(L^{p}\)-realization of the underlying linear operator, which allows us to make use of maximal regularity techniques. Of particular interest in this context is the identification of the domain and its fractional domain spaces, and the regularity of the semiflow. We conclude with an introduction of random dynamical systems and their random attracators. To apply this framework to SPDEs with dynamical boundary conditions is ongoing research, and we illustrate some of its key points.

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On Young regimes for locally monotone SPDEs

Florian Bechthold (University of Bielefeld, Germany)
Abstract: We consider the following SPDE on a Gelfand-triple \((V,H,V^{*})\):
\begin{equation*}
\begin{aligned}
du(t) &= A(t,u(t))dt + dI_{t}(u),\\
u(0) &= u_{0} \in H.
\end{aligned}
\end{equation*}
Given certain local monotonicity, continuity, coercivity and growth conditions of the operator \(A:[0,T]\times V\to V^{*}\) and a sufficiently regular operator \(I\) we establish global existence of weak solutions. In analogy to the Young regime for SDEs, no probabilistic structure is required in our analysis, which is based on a careful combination of monotone operator theory and the recently developed Besov rough analysis by Friz and Seeger. Due to the abstract nature of our approach, it applies to various examples of monotone and locally monotone operators \(A\), such as the \(p\)-Laplace operator, the porous medium operator, and an operator that arises in the context of shear-thickening fluids; and opreators \(I\), including additive Young drivers \(I_{t}(u) = Z_{t} - Z_{0}\), abstract Young integrals \(I_{t}(u) = \displaystyle\int_{0}^{t}\sigma(u_{s})\,dX_{s}\), and translated integrals \(I_{t}(u) = \displaystyle\int_{0}^{t}b(u_{s}-w_{s})\,ds\) that arise in the context of regularization by noise. In each of the latter cases, we identify corresponding noise regimes (i.e. Young regimes) that assure our abstract result to be applicable. In the case of additive drivers, we identify the Brownian setting as borderline, i.e. noises which enjoy slightly more temporal regularity are amenable to our completely pathwise analysis.

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Nonlocal equations with degenerate weights

Linus Behn (University of Bielefeld, Germany)
Abstract: We introduce fractional weighted Sobolev spaces with dgenerate weights. For these spaces we provide embeddings and Poincaré inequalities. When the order of fractional differentiability goes to \(0\) or \(1\), we recover the weighted Lebesgue and Sobolev spaces with Muckenhoupt weights, respectively. Moverover, we prove interior Hölder continuity and Harnack inequalities for solutions to the corresponding weighted nonlocal integro-differential equations. This naturally extends a classical result by Fabes, Kenig, and Serapioni to the nonlinear, nonlocal setting.

Joint work with Lars Diening (Bielefeld), Jihoon Ok (Seoul), and Julian Rolfes (Bielefeld).

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The Dirichlet problem and boundary regularity for nonlinear parabolic equations

Anders Björn (Linköping University, Sweden)
Abstract: The \(p\)-parabolic equation
\begin{equation}
\begin{aligned}
\partial_{t}u &= \Delta_{p}u := \text{div}\left( |\nabla u|^{p-2}\nabla u\right),
\end{aligned}
\end{equation}
is a nonlinear cousin of the classical heat equation. As such, it offers both difficulties and advantages compared with the heat equation. In the talk, we consider the Perron method for solving the Dirichlet problem for the \(p\)-parabolic equation in general bounded domains in \(\mathbb{R}^{n+1}\). Compared to space-time cylinders, such domains allow the space domain to change in time. Of particular interest will be boundary regularity for such domains, i.e. whether solutions attain their boundary data in a continuous way. Relations between regular boundary points and barriers will be discussed, as well as some peculiar examples and surprising phenomena related to boundary regularity.

Towards the end, if time permits, I will discuss the same type of questions for two other nonlinear cousins of the heat equation, the porous medium equation,
\begin{equation}
\begin{aligned}
\partial_{t}u &= \text{div}\left(u^{m}\right),
\end{aligned}
\end{equation}
and the so-called normalized \(p\)-parabolic equation,
\begin{equation}
\begin{aligned}
\partial_{t}u &=|\nabla u|^{2-p}\Delta_{p}u.
\end{aligned}
\end{equation}

This talk is based on collaborations with Jana Björn (Linköping), Ugo Gianazza (Pavia), Mikko Parviainen (Jyväsylä) and Juhana Siljander (Jyväskylä).

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Perron method and boundary regularity for nonlinear nonlocal problems

Jana Björn (Linköping Unviersity, Sweden)
Abstract: Consider a nonlocal Dirichlet problem for nonlinear operators of \(s\)-fractional \(p\)-Laplace type, such as \((-\Delta_{p})^{s}u = 0\), on a general bounded open set \(\Omega\). We define Perron solutions for arbitrary exterior Dirichlet data on the complement of \(\Omega\) and study when the upper and the lower Perron solutions coincide. Our definition of Perron solutions generalizes two earlier definitions and leads to some useful properties for the Perron solutions including uniqueness results and perturbations on sets of zero capacity. Perron solutions appear also in a characterization of regular boundary points. Finally, regular boundary points and sets of zero capacity will be compared for different values of \(p\) and \(s\), including the local case \(s=1\).

Joint work with Anders Björn and Minhyun Kim.

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Existence and boundedness of solutions to singular anisotropic elliptic equations

Barbara Brandolini (University of Palermo, Italy)
Abstract: In this talk, we discuss some recent results on the existence and uniform boundedness of solutions for a general class of Dirichlet anisotropic elliptic problems of the form
\begin{equation*}
\begin{aligned}
-\Delta_{\overrightarrow{p}}u+\Phi_{0}(u,\nabla u) &= \Psi(u,\nabla u) +f& &\text{in } \Omega,\\
u &= 0& &\text{on }\partial \Omega,
\end{aligned}
\end{equation*}
where \(\Omega\) is a bounded open subset of \(\mathbb{R}^N\) \((N\geq 2)\),
\begin{equation*}
\begin{aligned}
\Delta_{\overrightarrow{p}}u &= \sum_{j=1}^{N} \partial_{j} (|\partial_{j}u|^{p_{j}-2}\partial_{j}u)
\end{aligned}
\end{equation*}
and
\begin{equation*}
\begin{aligned}
\Phi_{0}(u,\nabla u) &= \left(\mathfrak{a}_{0}+\sum_{j=1}^{N} \mathfrak{a}_{j} |\partial_{j} u|^{p_{j}}\right)|u|^{m-2}u,
\end{aligned}
\end{equation*}
with \(\mathfrak{a}_{0}>0\), \(m,p_j>1\), \(\mathfrak{a}_{j}\geq 0\) for \(1\leq j\leq N\) and \(\frac{N}{p}=\sum_{k=1}^{N} (\frac{1}{p_{k}})>1\). We assume that \(f \in L^{r}(\Omega)\) with \(r>\frac{N}{p}\). The feature of this study is the inclusion of a possibly singular gradient-dependent term
\begin{equation*}
\begin{aligned}
\Psi(u,\nabla u) &= \sum_{j=1}^{N} |u|^{\theta_{j}-2}u\, |\partial_{j} u|^{q_{j}},
\end{aligned}
\end{equation*}
where \(\theta_{j}>0\) and \(0\leq q_{j}<p_{j}\) for \(1\leq j\leq N\).  

Based on joint works with Florica C. Cîrstea (The University of Sydney, Australia).

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\(s\)-minimal functions: existence and continuity

Claudia Bucur (University of Milano, Italy)
Abstract: We discuss a nonlocal fractional problem that serves as a nonlocal counterpart of the classical problem of functions of least gradient. We show how we obtain the existence of minimizers \(-\) called \(s\)-minimal functions \(-\) by leveraging their connection to nonlocal minimal sets. Additionally, we discuss the contuity of these minimizers and a weak formulation of the problem.

The results presented are obtained in collaboration with S. Dipierro, L. Lombardini, J. Mazón and E. Valdinoci.

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Complex interpolation and a new embedding for exponentially weighted modulation spaces

Leonid Chaichenets (Karlsruhe Institute of Technology, Germany)
Abstract: We recall the complex interpolation functor, introduce the notion of common retraction and coretraction for families of Banach spaces, and formulate a framework for identifying complex interpolation spaces. The fact that the complex interpolation space of a pair of Besov or modulation spaces is another Besov or modulation space, respectively, is easily recovered in this framework. Moreover, our result is applicable to modulation spaces with exponential regularity \(E_{p,q}^{s}\). In order to do so, we construct a common ambient space for the whole family of \(E_{p,q}^{s}\) and obtain the embedding \(E_{p_{1},q}^{s}\hookrightarrow E_{p_{2},q}^{s}\) for \(p_{1}\leq p_{2}\).

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Boundary value problems with rough boundary data\(\)

Robert Denk (University of Konstanz, Germany)
Abstract: Motivated by problems with boundary noise or with dynamic conditions on the boundary, we study boundary value problems where the data on the boundary does not belong to the classical trace space. To obtain unique solvability, one can consider a class of Sobolev spaces with anisotropic structure and obtain continuity of the trace with values in Besov spaces of negative order. In this way, we obtain unique solvability for problems with rough boundary data in the half-space and, to some extent, in domains. The results can be applied to prove the generation of an analytic semigroup for problems with dynamic boundary conditions.

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Symmetrization results for general nonlocal linear elliptic and parabolic problems

Vincenzo Ferone (University of Naples, Federico II, Italy)
Abstract: We discuss a Talenti-type symmetrization result in the form of mass concentration (i.e. integral comparison) for very general linear nonlocal elliptic problems, equipped with homogeneous Dirichlet boundary conditions. In this framework, the relevant concentration comparison for the classical fractional Laplacian can be reviewed as a special case of our main result, thus generalizing previous results obtained in collaboration with B. Volzone. Also a Cauchy-Dirichlet nonlocal linear parabolic problem is considered.

The results are contained in a joint paper with G. Piscitelli and B. Volzone.

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Gradient flow technique for porous medium equation driven by Lévy operators

Guy Fabrice Foghem Gounoue (TU Dresden, Germany)
Abstract: We examine a nonlocal porous medium equation, \(\partial_{t}u + \text{div}\left(u \nabla L^{-1}u\right) = 0\), where the pressure is governed by a general symmetric Lévy operator, \(L\), which is widely recognized as the generator of a symmetric Lévy stochastic process. A symmetric Lévy operator generalizes the classical fractional Laplace operator \(-\Delta^{s}\), with \(s\in(0,1)\). We construct weak solutions in the context of the corresponding nonlocal Sobolev space using the Jordan-Kinderlehrer-Otto (JKO) minimizing movement scheme. The absence of interpolation and various tools from classical fractional Sobolev spaces intensifies the complexity of our approach, requiring the exploration of Bernstein functions. Futhermore, we investigate the nonlocal-to-local convergence of the aforementioned problem.

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Asymptotics for eigenfunctions of the Laplacian in domains with small holes

Massimo Grossi (Sapienza University of Rome, Italy)
Abstract: Let us consider the following eigenvalue problem in \(\mathbb{R}^{N}\), \(N\ge 2\),
\begin{equation}
\begin{aligned}
-\Delta u_{\epsilon} &= \lambda_{\epsilon}u_{\epsilon} & & \text{in }\Omega_{\epsilon} = \Omega\setminus B(P,\epsilon),\\
u_{\epsilon} &= 0 & & \text{on }\partial\Omega_{\epsilon},
\end{aligned}
\end{equation}
where \(B(P,\epsilon)\) is the ball centered at \(P\in\Omega\) and small radius \(\epsilon\). It is well known that the previous problem converges as \(\epsilon\to 0\) in a suitable sense to
\begin{equation}
\begin{aligned}
-\Delta u &= \lambda u & & \text{in }\Omega,\\
u &= 0 & & \text{on }\partial\Omega.
\end{aligned}
\end{equation}
In this talk we want to give additional information on the convergence of the pair \((\lambda_{\epsilon},u_{\epsilon})\) to \((\lambda, u)\) as \(\epsilon\to 0\). In particular we try to give an answer to the following questions,

• If \(u(P)\ne 0\) we have that \(u_{\epsilon}\) does not converge uniformly to \(u\) near \(\partial\Omega_{\epsilon}\). Is it possible to describe more precisely the behavior of \(u_{\epsilon}\) near \(\partial\Omega_{\epsilon}\)?
• If \(\lambda\) is a multiple eigenvalue is it true that the eigenvalues \(\lambda_{\epsilon}\) are simple (Uhlenbeck's property) for \(\epsilon\) small?
• What about the nodal region of \(u_{\epsilon}\) near \(\partial B(P,\epsilon)\)?

This is a joint paper with Laura Abatangelo (Politecnico di Milano, Italy).

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Boundary regularity and Hopf lemma for stable operators in \(C^{1,\text{dini}}\)-domains (Poster)

Florian Grube (University of Bielefeld, Germany)
Abstract: We prove sharp boundary Hölder regularity for solutions to equations involving stable integro-differential operators in \(C^{1,\text{dini}}\)-domains. This result is new even for the fractional Laplacian. A Hopf-type boundary lemma is proven, too. One novelty of this work is that the regularity estimate is robust as \(s\to 1^{-}\) and we recover the classical results for second order equations.

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Strong solutions to a class of degenerate McKean-Vlasov SDEs with coefficients of Nemytskii-type

Sebastian Grube (University of Bielefeld, Germany)
Abstract: While the nondegenerate case is well-knwon, there are only few results on the xistence of strong solutions to McKean-Vlasov SDEs with coefficients of Nemytskii-type in the degenerate case. We consider a broad class of degenerate nonlinear Fokker-Planck(-Kolmogorov) equations with coefficients of Nemytskii-type. This includes, in particular, the classical porous medium equation perturbed by a first-order term with initial datum in a subset of probability densities, which is dense with respect to the topology inherited from \(L^{1}\), and, in the on-dimensional setting, the classical porous medium equation with initial datum in an arbitrary point \(x_{0}\) in the real numbers. For these kind of equations the existence of a Schwartz-distributional solution \(u\) is well-known. We show that there exists a unique strong solution to the associated degenerate McKean-Vlasov SDE with time marginal law densities \(u\). In particular, every weak solution to this equation with time marginal law densities \(u\) can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional yields a weak solution with time marginal law densities \(u\).

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Lyapunov couples and regularisation of semigroups

Jan Hausmann (TU Dresden, Germany)
Abstract: In this talk we will consider functional inequalities such as
\begin{equation*}
\begin{aligned}
\varphi(T_{t}u) + \int_{0}^{t}\psi(T_{s}u)\,ds &\leq \varphi(u),
\end{aligned}
\end{equation*}
where \((T_{t})_{t\geq 0}\) denotes a (nonlinear) semigroup on a closed subset \(D\) of a Banach space \(X\) and \(\varphi,\psi:X\to\mathbb{R}\cup\{\infty\}\). Here, we call a pair \((\varphi,\psi)\) Lyapunov couple for \((T_{t})_{t\geq 0}\) if the above inequality is satisfied for all \((u,t)\in D\times[0,\infty)\). Using Sobolev type inequalities, we will see examples of Lyapunov couples for the semigroup \((e^{-t\Delta_{p}^{D}})_{t\geq 0}\) generated by the Dirichlet \(p\)-Laplacian, \(\Delta_{p}^{D}\). In particular, we will use Lyapunov couples to derive a regularisation effect of the form
\begin{equation*}
\begin{aligned}
\varphi(T_{t}u) &\lesssim t^{-1}\varphi(u).
\end{aligned}
\end{equation*}

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Boundary regularity results for the fractional \(p\)-Laplacian

Antonio Iannizzotto (University of Cagliari, Italy)
Abstract: I will present some results about global regularity for weak solutions of inhomogeneous elliptic equations drive by the \(s\)-fractional \(p\)-Laplacian, with zero Dirichlet conditions, deriving from old and new collaborations with S. Mosconi and M. Squassina. Precisely, it will be seen that any sch solutions is Hölder continuous up to the boundary with the optimal exponent \(s\), and that the same solution divided by the \(s\)-power of the distance from the boundary is Hölder continuous as well, thus providing a fractional order analogue of Lieberman's global gradient regularity result for the classical \(p\)-Laplacian.

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Optimal boundary regularity and Green function estimates

Minhyun Kim (Hanyang University, South Korea)
Abstract: We study the \(C^{s}\) boundary regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in \(C^{1,\alpha}\) domains. As an application of our results, we establish sharp two-sided Green function estimates in \(C^{1,\alpha}\) domains.

This talk is based on joint work with Marvin Weidner.

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Liouville's theorems for Lévy operators

Mateusz Kwaśnicki (Wroław University of Science and Technology, Poland)
Abstract: Classical Liouville's theorem states that every bounded (or merely positive) harmonic function is constant. Recently a number of authors studied a similar question for solutions of the more general equation \(Lu = 0\), where \(L\) is a Lévy operator: a translation-invariant (or constant-coefficient) non-local ''elliptic'' operator on the Euclidean space. Chan-D'Ambrosio-Li and by Fall studied the fractional Laplace operator, Alibaud-del Teso-Endal-Jakobsen proved the theorem for bounded solutions and general Lévy operators, Berger-Schilling gave an independent and simplified proof together with partial extension to positive solutions, and there are related works by Barlow-Bass-Gui, Beger-Schilling-Shargorodsky, Fall-Weth, and Kühn. Together with Tomasz Grzywny we were able to prove a general Liouville's theorem for positive solutions and general Lévy operators, as well as a number of variants of the theorem for unbounded signed solutions. We also constructed a counterexample which shows that with no furhter assumption, Liouville's theorem for unbounded signed solutions does not hold. During my talk I will discuss the history of the problem and our contribution, which can be found at arXiv:2301.08540.

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Nonlocal Meyers' example

Ho-Sik Lee (University of Bielefeld, Germany)
Abstract: We consider examples of solutions to the nonlocal equation with irregular coefficients, which tell us that the higher differentiability and integrability of solutions are limited. Inspired by Meyers' example given in [Meyers '63] and [Kh. Balci, Diening, Giova, Passarelli di Napoli '22], we give nonlocal versions of such examples that are also robust when the order of the nonlocal equation converges to \(2\).

This is the joint work with Prof. Anna Kh. Balci (Charles), Prof. Lars Diening (Bielefeld), and Prof. Moritz Kassmann (Bielefeld).

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Capacity Theory for nonlinear equations and its application

Ki-Ahm Lee (Seoul National University, South Korea)
Abstract: In this talk we would like to discuss how capacity theory for nonlinear local or nonlocal operators shows up in applications: Homogenization for the highly oscillating obstacles and Wiener Criterion for the boundary value problems. At Homogenization theory, we will consider the influence of the capacity of periodically or randomly oscillating obstacles for various nonlinear operators to find the effective equations. For Wiener Criterion for the boundary value problems, we will measure the influence of the capacity of the complement of the domains in shrinking local regions to Weak Harnack Inequalities.

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Singularities of solutions of nonlocal nonlinear equations

Se-Chan Lee (Korea Institute for Advanced Study, South Korea)
Abstract: In this talk, we study the local behaviors of weak solutions, with possible singularities, of nonlocal nonlinear equations. We first prove that sets of capacity zero are removable for harmonic functions under certain integrability conditions. By adopting this removability theorem, we characterize the behavior of singular solutions near an isolated singularity. Our approach relies on the substantial use of local estimates such as Caccioppoli estimates and the Harnack inequality, together with the nonlocal nonlinear potential theory.

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Schrödinger systems in \(L^{p}\) spaces: Maximal regularity and generation results (Poster)

Vincenzo Leone (University of Salerno, Italy)
Abstract: In this work, we will consider the vector-valued Schrödinger operator \(-\Delta + V\) in \(L^{p}(\mathbb{R}^{d},\mathbb{R}^{m})\), where the potential term \(V\) is a non-negative definite and symmetric matrix-valued function. Under suitable assumption on the potential, we obtain maximal inequality in \(L^{1}(\mathbb{R}^{d},\mathbb{R}^{m})\). Assuming in addition that the minimal eigenvalue of \(V\) belongs to some reverse Hölder class, we also obtain maximal inequalities in \(L^{p}(\mathbb{R}^{d},\mathbb{R}^{m})\), for a class of \(p\). An application of these results in the context of generation of \(C_{0}\)-semigroup is also given.

Work in collaboration with Davide Addona, Luca Lorenzi and Abdelaziz Rhandi.

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Sharp constant and extremal function in a weighted trace-Sobolev inequality (Poster)

Jingjing Ma (Shaanxi Normal University, China)
Abstract: In this work, we show the Sharp constant in trace-Sobolev inequality by the method of mass transportation.

This is joint work with Daniel Hauer.

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Singular domain perturbation for nonlocal elliptic operators (Poster)

Kyle McLaren (The University of Sydney, Australia)
Abstract: In this research project, I am to adapt the local theory of singular domain perturbation established by Dancer and Daners [J. Differential Equations, 138 (1997)] and Daners [Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002)], [Handbook of differential equations: stationary partial differential equations, Vol. VI, 2008] to the nonlocal setting. We start by proving new uniform Sobolev embeddings for fractional Sobolev spaces on domains with shrinking holes to a singular set. As an application of this, we aim to show \(L^{p}\)-convergence \((1\leq p<\infty)\) of weak solutions to elliptic boundary value problems governed by the fractional Laplacian where the solutions satisfy (nonlocal) Neumann/Robin type boundary conditions.

This research project is the topic of my masters thesis at The University of Sydney under the supervision of Daniel Hauer.

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Weak solutions to gradient flows in metric measure spaces

José M. Mazón (University of Valencia, Spain)
Abstract: Analysis of evolution equations in metric measure spaces requires very different methods from the Euclidean setting. The reason is that, in general, there is no notion of a gradient of a function. Instead, one can use the minimal upper gradient, which roughly corresponds to the length of the gradient. Here, we show how to introduce the notion of weak solutions in metric measure spaces in the model case of the -Laplacian evolution equation (including the borderline case \(p = 1\), i.e., the total variation flow). For \(p > 1\) it has been previously studied as the gradient flow in \(L^{2}\) of the p-Cheeger energy. Using the first-order differential structure on a metric measure space introduced by Gigli, we characterise the subdifferential in \(L^2\) of the \(p\)-Cheeger energy. This leads to a new definition of solutions to the \(p\)-Laplacian evolution equation in metric measure spaces, in which the gradient is replaced by a vector field (defined via Gigli’s differential structure) satisfying some compatibility conditions. Existence of solutions in this sense is obtained indirectly, using the Fenchel-Rockafellar duality theorem. For \(p = 1\), we additionally encounter the problems known already in the Euclidean case, i.e. that the solutions for any given time do not lie in any Sobolev space, but only in the space of functions of bounded variation. For this reason, we need to extend Gigli’s differential structure to this case, and prove a Green-Gauss formula similar to the one by Anzellotti for Euclidean spaces. Then, we use it to characterise the \(1\)-Laplacian operator and introduce weak solutions to the total variation flow.

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Evolution equations with eventually positive solutions (Poster)

Jonathan Mui (University of Wuppertal, Germany)
Abstract: This poster is an invitation to the topic of eventual positivity for evolution equations, and highlights both the abstract aspects of the theory alongside concrete PDE applications. While evolution equations with positivity preserving properties are well-known and have been studied for a long time, a systematic theory for the more subtle property of eventual positivity has only been developed recently. On one hand, the theory draws on many established techniques from the theory of positive operator semigroups, and on the other hand, takes inspiration from results about positivity properties of higher-order elliptic equations.

This work is part of the ongoing DFG project 515394002.

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Nonlinear nonlocal potential theory at the gradient level

Simon Nowak (University of Bielefeld, Germany)
Abstract: We present pointwise gradient potential estimates for a class of nonlinear nonlocal equations related to quadratic nonlocal energy functionals. Our pointwise estimates imply that the first-order regularity properties of such general nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian.

The talk is based on joint work with Lars Diening, Kyeongbae Kim and Ho-Sik Lee.

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On some properties of Steklov eigenfunctions

Angela Pistoia (Sapienza University of Rome, Italy)
Abstract: We focus on a couple of properties of the eigenfunctions of Steklov problem on a compact Riemannian manifold with boundary.

First, we give a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of the manifold and of the number of its sign changes the boundary. Based on a joint work with Luca Battaglia (Università degli Studi Roma Tre) and Luigi Provenzano (Sapienza Università di Roma)

Next, we disprove the conjectured validity of Courant’s theorem for the traces of Steklov eigenfunctions building a Riemannian metric for which the n-th eigenfunction has an arbitrary number of nodal domains on the boundary. Based on a joint work with Alberto Enciso (ICMAT Madrid) and Luigi Provenzano (Sapienza Università di Roma).

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Kernel estimates for parabolic systems of PDEs with unbounded coefficients

Marianna Porfido (TU Bergakademie Freiberg, Germany)
Abstract: In this talk we consider a class of systems of nondegenerate elliptic partial differential equations with unbounded coefficients with possibly unbounded diffusion coefficients which may vary equation by equation. In particular, we deal with a vector-valued elliptic operator \(\mathcal{A}\) in divergence form defined on smooth functions \(f:\mathbb{R}^{d}\to\mathbb{R}^{m}\) by
\begin{aligned}
(\mathcal{A}f)_{h} &= \text{div}(Q^{h}\nabla f_{h}) + \langle b^{h},\nabla f_{h} \rangle - (Vf)_{h},
\end{aligned}
for \(h = 1,\ldots, m\), where \( Q^{h}:\mathbb{R}^{d}\to \mathbb{R}^{d\times d}\), \(b^{h}:\mathbb{R}^{d}\to\mathbb{R}^{d}\) for every \(h=1,\ldots, m\) and \(V:\mathbb{R}^{d}\to\mathbb{R}^{m\times m}\). Under suitable assumptions, we prove pointwise upper bounds for the transition kernels of the semigroup associated in \(C_{b}(\mathbb{R}^{d};\mathbb{R}^{m})\) with the operator \(\mathbf{\mathcal{A}}\). The idea is to adapt and generalize to our setting the techniques exploited in the scalar case based on time-dependent Lyapunov functions for the parabolic operator \(D_{t}+\mathbf{\mathcal{A}}\). We finally illustrate our results in case of polynomially and exponentially growing coefficients.

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Positivity and asymptotic behaviour of solutions to a generalized nonlocal fast diffusion equation

Fernando Quirós (Autonomous University of Madrid, Spain)
Abstract: We study the positivity and asymptotic behaviour of nonnegative solutions of a general nonlocal fast diffusion equation, \(\partial_t u + \mathcal{L}\varphi(u) = 0,\) and the interplay between these two properties. Here \(\mathcal{L}\) is a stable-like operator and \(\varphi\) is a singular nonlinearity. We start by analysing positivity by means of a weak Harnack inequality satisfied by a related elliptic (nonlocal) equation. Then we use this positivity to establish the asymptotic behaviour: under certain hypotheses on the nonlocal operator and nonlinearity, our solutions behave asymptotically as the Barenblatt solution of the standard fractional fast diffusion equation. The main difficulty stems from the generality of the operator, which does not allow the use of the methods that were available for the fractional Laplacian. Our results are new even in the case where \(\varphi\) is a power.

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Kernel estimates for a class of fractional Kolmogorov operators

Abdelaziz Rhandi (University of Salerno, Italy)
Abstract: In this talk, we consider a measure space \((X,\mu)\) with \(\sigma\)-finite measure \(\mu\) and a non-negative self-adjoint operator \(A\) on \(L^{2}(\mu)\) such that \(-A\) generates a symmetric Markov semigroup. In addition to the semigroup property and the strong continuity, a symmetric Markov semigroup is a family of positive preserving operators acting on bounded measurable functions, which preserve constant functions and are symmetric on \(L^{2}\). We prove pointwise bounds for the kernel associated to the fractional operator \(-A^{\alpha}\) for \(0<\alpha<1\). The main tools are weighted Nash inequalities. Finally, we illustrate our results in concrete examples.

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The Dirichlet problem for the one-dimensional Rudin-Osher-Fatemi functional

Piotr Rybka (Univeristy of Warsaw, Poland)
Abstract: We study minimizers of the following functional,
\begin{aligned}
\Phi_{f,\lambda,\varphi}(u) &=
\begin{cases}
\displaystyle\int_{0}^{L}|Du|\,dx + \frac{\lambda}{2}\int_{0}^{L}(u-f)^{2}\,dx, \quad\gamma u = \varphi, &u\in BV(\Omega),\\
+\infty, &\text{otherwise},
\end{cases}
\end{aligned}
where \(\Omega=(0,L)\) and \(\gamma:BV(\Omega)\to L^{1}(\partial\Omega)\equiv \mathbb{R}^{2}\) is the trace operator. The problem we face is the lack of lower semicontinuity of \(\Phi_{f,\lambda,\varphi}\). As a result minimizers need not exist.

We offer a few sets of sufficient conditions for the existence of minizers of \(\Phi_{f,\lambda,\varphi}\) satisfying the Dirichlet boundary conditions in the trace sense. In one case we rely on the methods specific for the total variation flow. We also present a few counter-examples.

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Concavity principles for the principal frequency in Gauss Space

Paolo Salani (Università degli Studi di Firenze, Italy)
Abstract: I will prove a Brunn-Minkowski inequality for the dirichlet eigenvalue of the Ornstein-Uhlenbeck operator and the log-concavity of the associated eigenfunction of a convex domain.

This talk is based on a joint work with A. Colesanti, E. Francini, G. Livshyts.

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Space-Time Analyticity in the Heston volatility model in Mathematical Finance

Peter Takáč (University of Rostock, Germany)
Abstract: We begin by a brief presentation of a well-known mathematical model for European option pricing in a market with stoachastic volatility: the popular heston volatility model (Rev. Financial Studies, 1993). European options are used for market completion. We explain the connection between a complete market and the analyticity of the weak solution to a general, strongly parabolic linaer Cauchy problem of second order in \(\mathbb{R}^{N}\times(0,T)\, (N=2)\) with analytic coefficients (in space and time variables). The analytic smoothing property is expressed in terms of holomorphic continuations of global (weak) \(L^{2}\)-type solutions to the system. Given \(0<\xi'<\infty\) and \(0<T'<T<\infty\), we sketch a proof that any \(L^{2}\)-type solution \(u:\mathbb{R}^{1}\times(0,\infty)\times(0,T)\subset\mathbb{R}^{2}\times(0,T)\to \mathbb{R}^{1},\, u\equiv u(x,v,t)\), possesses a bounded holomorphic continuation \(u(x+iy,\xi+i\eta,\sigma+i\tau)\) into a complex domain in \(\mathbb{C}^{N}\times\mathbb{C}\,(N=2)\) defined by \((x,\xi,\sigma)\in\mathbb{R}^{1}\times(\xi',\infty)\times(T',T),\, |y|<A'_{1},\, |y|<A'_{2}\), and \(|\tau|<B'\), where \(A'_{1},A'_{2},B'>0\) are constants depending upon \(\xi'\) and \(T'\). The proof uses the extension of a solution to an \(L^{2}\)-type solution in a complex domain in \(\mathbb{C}^{2}\times\mathbb{C}\), such that this extension satisfies the Cauchy-Riemann equations. The holomorphic extension is thus obtained in a (weighted) Hardy space \(H^{2}\). A serious difficulty in the Heston model is that the solution is sought only in a half-space \(\mathbb{H}=\mathbb{R}^{1}\times(0,\infty)\) in \(\mathbb{R}^{2}\) with rather complicated dynamic boundary conditions at the boundary \(\partial\mathbb{H}=\mathbb{R}^{1}\times\{0\}\); a similarity with the Feller boundary condition (Ann. Math., 1951) will be discussed. We avoid this trouble by a suitable choice of the weight in the weighted \(L^{2}\) space.

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