Advances in Nonlinear Elliptic and Parabolic PDEs

Invited speakers

Organizers

• Piotr Biler (piotr.biler@math.uni.wroc.pl)
• Matteo Bonforte (matteo.bonforte@uam.es)
• Miłosz Krupski (milosz.krupski@math.uni.wroc.pl)
• Bruno Volzone (bruno.volzone@uniparthenope.it)

Schedule

Abstracts

Pedro Aceves-Sanchez

Fractional diffusion limit of a linear kinetic transport equation in a bounded domain

In recent years, the study of evolution equations featuring a fractional Laplacian has received many attention due the fact that they have been successfully applied into the modelling of a wide variety of phenomena, ranging from biology, physics to finance. The stochastic process behind fractional operators is linked, in the whole space, to an \(\alpha\)-stable processes as opposed to the Laplacian operator which is linked to a Brownian stochastic process.

In addition, evolution equations involving fractional Laplacians offer new interesting and very challenging mathematical problems. There are several equivalent definitions of the fractional Laplacian in the whole domain, however, in a bounded domain there are several options depending on the stochastic process considered.

In this talk we shall present results on the rigorous passage from a velocity jumping stochastic process in a bounded domain to a macroscopic evolution equation featuring a fractional Laplace operator. More precisely, we shall consider the long-time/small mean-free path asymptotic behaviour of the solutions of a re-scaled linear kinetic transport equation in a smooth bounded domain.

Begoña Barrios Barrera

Periodic solutions for the one-dimensional fractional Laplacian

Along this talk we establish the existence of periodic solutions of the nonlocal problem \((-\Delta)^s u= f(u)\) in \(\mathbb{R}\), where \((-\Delta)^s\) stands for the \(s\)-Laplacian, \(s\in (0,1)\). We introduce a suitable framework which allows linking the searching of such solutions into the existence of the ones of a semilinear problem in a suitable Hilbert space. Then with the usual tools of nonlinear analysis, we get the existence theorems which are lately enlightened with the analysis of some examples like the Benjamin-Ono equation.

Claudia Bucur

Behavior of nonlocal minimal surfaces for small values of the fractional parameter

Nonlocal minimal surfaces are defined as boundary of sets that minimize the fractional perimeter in a bounded open set \(\Omega\subset \mathbb R^n\) among all sets with fixed data outside of \(\Omega\). It is known that if the fixed exterior data is a half-space, the only \(s\)-minimal set is the half-space. On the other hand, if one removes (even from far away) some small set from the half-space, for \(s\) small enough the \(s\)-minimal set completely sticks to the boundary, that is, the \(s\)-minimal set is empty inside \(\Omega\). Starting from this example, in this talk we present the behavior of \(s\)-minimal surfaces when the fractional parameter is small. We classify the behavior of \(s\)-minimal surfaces with respect to the fixed exterior data. So, for \(s\) small and depending on the data at "infinity", the \(s\)-minimal set can be either empty in \(\Omega\), fill all \(\Omega\), or possibly develop a wildly oscillating boundary.

Jan Burczak

Fractional Patlak-Keller-Segel system

Consider the following fractional generalization of the classical Patlak-Keller-Segel equation \[ u_t +{ (- \Delta)^\frac{\alpha}{2}} u = - \chi {{\rm div} \big( u \nabla (- \Delta)^{-1}(u - \langle u \rangle) \big) } + f(u), \] motivated by 'Levý flight foraging hypothesis', widely present in biology.

I will discuss a series of regularity results, obtained in collaboration with Rafael Granero-Belinchón, including

1. in case of 'critical' dissipation: A disproof of a finite-time blowup conjecture for \(f (s) \equiv 0\), \(\alpha=1\) on a circle
2. in case of a weaker than 'critical' dissipation, but with a damping: classical solvability and boundedness of solutions for \(f (s) \equiv rs (1-s)\) on \(d\)-dimensional torus

Daniele Castorina

Ancient solutions of superlinear heat equations on Riemannian manifolds

We study the qualitative properties of ancient solutions of superlinear heat equations in a Riemannian manifold, with particular attention to positivity and triviality in space.

This is joint work with Carlo Mantegazza (Napoli Federico II)

Azahara DelaTorre

On higher dimensional singularities for the fractional Yamabe problem

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

This is a work done in collaboration with Weiwei Ao, Hardy Chan, Marco Fontelos, Manuel del Pino, María del Mar González and Juncheng Wei.

Félix del Teso

Theoretical and numerical aspects for nonlocal (and local) equations of porous medium type

We consider the following problem of porous medium type: \[ \left\{ \begin{split} &\partial_t u(x,t)-(L^\sigma +\mathcal{L}^{\mu})\left[\varphi(u)\right](x,t)=f(x,t), \qquad && (x,t)\in \mathbb{R}^N\times (0,\infty),\\ &u(x,t)=u_0(x),&& x\in \mathbb{R}^N, \end{split} \right. \] where \(\varphi: \mathbb{R}^N\to\mathbb{R}\) is continuous and nondecreasing, and \[ \begin{split} L^\sigma[v](x)&=\text{Tr}\left(\sigma\sigma^T D^2v(x)\right), &&\textbf{ (local diffusion)}\\ \mathcal{L}^\mu[v](x)&=\text{P.V.} \int_{|z|>0}\left(v(x+z)-v(x)\right)d\mu(z), &&\textbf{ (nonlocal diffusion)} \end{split} \] with \(\sigma\in \mathbb{R}^{N\times p}\) and \(\mu\) symmetric measure s.t. \(\int \min\{|z|^2,1\}d\mu(z)<+\infty\).

We will present a general overview of some of the results obtained in collaboration with J. Endal and E.R. Jakobsen:

• Uniqueness of distributional solutions.
• Continuous dependence on \(L^\sigma +\mathcal{L}^{\mu}\), \(\varphi\) and \(u_0\).
• Unified theory of monotone numerical schemes of finite difference type. Here we use the fact that operators in the class of \(\mathcal{L}^\mu\) includes discretizations of \(L^\sigma+\mathcal{L}^\mu\). This fact allows us to use a pure PDE approach.
• We also propose a branch of discretizations and schemes and analyze their accuracy.
• As a consequence of numerics, we obtain existence of distributional solutions together with interesting properties like \(L^1\)-contraction, \(C([0,T],L^1_{\text{loc}}(\mathbb{R}^N))\) regularity, energy estimates, ...

Filomena Feo

Long-time asymptotics for nonlocal porous medium equation with absorption or convection

A large variety of models for conserved quantities in continuum mechanics or physics are described by the continuity equation \( u_\tau + \nabla \cdot (u\mathbf{v})=0,\) where the density distribution \(u(y,\tau)\) evolves in time \(\tau\) following a velocity field \(\mathbf{v}(y,\tau)\). According to Darcy's law, the velocity \(\mathbf{v}\) is usually derived from a potential \(p\) in the form \(\mathbf{v} =-\mathcal{D}\nabla p\) for some tensor \(\mathcal{D}\). In porous media, the power-law relation \(p=u^m\) is commonly proposed. Although local constitutive relations like \(p=u^m\) were successful in numerous practical models, there are situations where the potential (or pressure) \(p\) depends non-locally on the density distribution \(u\). The simplest prototypical example is \(p=(-\Delta)^{-s}u\), expressed as the Riesz potential of \(u\). The resulting evolution equation then becomes \[ u_\tau - \nabla\cdot(u\nabla(-\Delta)^{-s}u)=0,\] and basic questions like existence, uniqueness and regularity of solutions have been studied thoroughly in papers by L. A. Caffarelli and J.L. Vázquez. While in general it is difficult to obtain quantitative properties of solutions to non-local nonlinear equations, this equation possesses special features that enable one to study the long term behaviours in terms of its self-similar solution. Using similarity variables motivated from the scaling relations, the transformed equation has an entropy function so that the convergence towards the self-similar profile in one dimension can be established by the well-known entropy method. In this talk, we consider two variants of this equation with an absorption term or a convection term.

This talk is based on a joint work with Y. Huang and B. Volzone.

Gabriele Grillo

Nonlinear diffusion on negatively curved manifolds

We report on some recent results on the asymptotic behaviour of solutions to the porous medium equation on negatively curved manifolds. The results depend crucially on the growth assumptions on curvature at infinity. In fact if curvature is superquadratic a suprising connection with an associated elliptic problem, which is shown to have a solution in that case, arise.

Espen R. Jakobsen

The Liouville theorem and linear operators satisfying the max principle: Classification in the constant coefficient case

I will present a classification of linear operators \(\mathcal L\) satisfying the Liouville theorem: Bounded solutions of \(\mathcal Lu=0\) are constant. Our results give necessary and sufficient conditions for all the generators of Levy processes, or in other words, the constant coefficient linear operators satisfying the max principle. Some examples of such generators are the Laplace, fractional Laplace, and discrete finite differences operators. The main novelty is the inclusion of the nonlocal part of such operators. Our proofs are short and natural and differs from most proofs e.g. for the fractional Laplacian.

This is joint work with Nathael Alibaud (Besancon, France), Felix del Teso, and Jørgen Endal (both NTNU, Norway)

Grzegorz Karch

On a nonlinear nonlocal diffusion equation

In the talk, I will describe an abstract framework for non-local non-linear diffusion, by which we mean a phenomenon with properties strongly associated to diffusive processes such as the conservation of mass, the maximum principle, and the comparison principle. This framework encompasses some of the known examples of equations like the fractional porous medium equation or the equation with the fractional \(p\)-Laplacian, but it also opens up the space for new examples to be constructed and studied.

Tomasz Klimsiak

Renormalized solutions of nonlocal semilinear equations with measure data

We generalize the notion of renormalized solution to semilinear elliptic and parabolic equations involving operator associated with general (possibly nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We show that under mild integrability assumption on the data a quasi-continuous (quasi-càdlàg) function \(u\) is a renormalized solution to an elliptic (or parabolic) equation iff \(u\) is its probabilistic solution, i.e. \(u\) can be represented by a suitable nonlinear Feynman-Kac formula. Next we present the existence results for a broad class of local and nonlocal semilinear equations.

Matteo Muratori

From stochastic completeness to nonlinear diffusions and back

We prove that the conservation of mass for the heat semigroup on a complete Riemannian manifold \(M\) (namely stochastic completeness), hence a linear property, is equivalent to uniqueness for nonlinear evolution equations of fast diffusion type on \(M\), in the class of nonnegative bounded solutions. This connection was well known only in the linear framework, that is for the heat equation itself. More precisely, we consider equations of the form \(u_t=\Delta \phi(u)\), where \(\phi\) is an arbitrary nonnegative, concave, increasing function, regular outside the origin and satisfying \(\phi(0)=0\). We stress that either property is also shown to be equivalent to nonexistence of nontrivial, nonnegative bounded solutions to the semilinear elliptic equation \(\Delta W=\phi^{-1}(W)\), with the same \(\phi\) as above. As a consequence, we provide sharp criteria for uniqueness or nonuniqueness of nonnegative bounded solutions to fast diffusion-type equations and existence or nonexistence of nontrivial, nonnegative bounded solutions to the associated semilinear elliptic equations on \(M\), which seem to be completely new in the literature. In particular, our results show that there is a wide class of manifolds where uniqueness of bounded solutions to the fast diffusion equation \(u_t = \Delta u^m \), with \( m \in (0,1) \), fails. This is in clear contrast with the Euclidean setting, in which uniqueness does hold for merely \(L^1_{\mathrm{loc}}\) solutions due to the seminal paper by Herrero and Pierre.

This is a joint work with G. Grillo and K. Ishige.

Maria Rosaria Posteraro

Stability of the log-Sobolev inequality for the Gaussian measure

We study the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension. We obtain a sharp lower bound using a distance introduced by Bucur and Fragalà. Thereafter, we investigate the stability issue with tools from Fourier analysis.

Fernando Quirós

Heat equations for anisotropic nonlocal operators with singular forcing

We prove existence, uniqueness and regularity for bounded weak solutions of a nonlocal heat equation associated to a stable diffusion operator. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions.

Joint work with Arturo de Pablo and Ana Rodríguez.

Antonio Segatti

Functional framework and topological defects in nematic shells

In this talk I will report on a series of joint results obtained in collaboration with Giacomo Canevari (Bilbao) and Marco Veneroni (Pavia) on nematic liquid crystals smeared on curved substrates (nematic shells). This structures offer an interesting playground where modelling, analysis of PDEs, Calculus of Variations, Topology and Geometry meet. In this talk I will first discuss how the topology of the shell influences the choice of a proper functional setting of the problem. Secondly, I will discuss how the defects emerge and I will present their energetics.

Vincenzo Vespri

Harnack at large for degenerate/singular operators

In his celebrated paper “ A Harnack Inequality for Parabolic Differential Equations, Comm. Pure Appl. Math., 17, (1964), 101 - 134” Moser, in Theorem 1, extended the Harnack inequality to linear parabolic operators with elliptic and measurable coefficients. Moser focussed his attention on pointwise estimates of the solutions. More specifically, in Theorem 2 and estimate (1.7), he proved that there exists a positive constant \(c\) such that, for any \(x\) and \(y\) in \(\mathbb{R}^N\), for any \(0 < s < t < T\) and for any nonnegative solution \(u\) of \[\partial_t u = \sum^N_{i, j=1} D_i \big( a_{i j}(x,t)D_ju \big)\] in \(\mathbb{R}^N\times(0,\infty)\), we have \[u(x,t) \geq u(y,s) \left(\frac{s}{t}\right)^c e^{ −c \left( 1+ \frac{|x−y|^2}{t−s}\right)}.\] Let us remark that these estimates, even if not optimal especially in time variable, give the idea of the strong connection between Harnack estimates and the well known exponential behavior of the fundamental solution. Moser proved this estimate by using a technique called Harnack chain which consists in iterating the Harnack estimates. However, it is known that this technique produces non optimal estimates. By using Nash techniques, many authors proved sharp estimates from above and from below for linear operators in different settings, cf. Li and Yau , Auscher and Coulhon , Grigor’yan and Telcs and Saloff-Coste. In “A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash Arch. Rat. Mech. Anal. 96 (1986), 327–338”, Fabes and Stroock proved that the Gaussian estimate are equivalent to a parabolic Harnack inequality. In his book, “Aspects of Sobolev-type inequalities (1992)”, Saloff-Coste gave a proof of Moser’s estimate with the right coefficients. In this talk we investigate the connection between the fundamental solution estimates and the parabolic Harnack inequality also in the case of quasilinear degenerate/singular parabolic operators.

Dariusz Wrzosek

The ability to move gives benefits in a two species competition model with nutrient taxis

In a joint work with Piotr Krzyżanowski (Warsaw) and Michael Winkler (Paderborn) a model describing the competition of two species for a common nutrient is studied. It is assumed that one of the competitors is motionless while the other has the ability to move upwards gradients of the nutrient density. It is proved that under suitable assumptions on the initial data, in the long time perspective the ability to move turns out to be a crucial feature providing competitive advantage irrespectively of a possible difference between the species with regard to their rates of nutrient uptake and proliferation.