 TALKS

....................................

Barnaś Sylwia

The existence of solutions for differential inclusions involving p(x)Laplacian with the signchanging weight


We consider a nonlinear elliptic differential inclusions with p(x)Laplacian and with Dirichlet boundary condition. These are the socalled hemivariational inequalities and they are derived with the help of subdifferential in the sense of Clarke.
We provide the necessary conditions for the existence
of a solution for some partial differential inequality in the situation when the weight can change the sign. It is a lot of papers with the negative weight. There are also some results in the situation when it is positive, but in some small intervals or with a lot of restrictions on exponent $p$ such as $p^+ < N$ or $\sqrt{2}p^ > N$. This assumptions are necessary to demonstrate the compactness of the embedding of the variable Sobolev space into the spaces $C^0 (\Omega)$ and $L^\infty (\Omega)$.
Now we significantly expand the class of considered functions because we claim that our weight can be from $\mathbb{R}^N$. By using the Ekeland variational principle and the properties of variational Sobolev spaces, we establish conditions which ensure the existence of a solution for our problem.

Benedikt Jiří

Estimates of the Principal Eigenvalue of the pLaplacian and the pbiharmonic Operator


We provide estimates from below and from above for the principal eigenvalue of the pLaplacian and the pbiharmonic Operator on a bounded domain. We apply these estimates to study the asymptotic behavior of the principal eigenvalue for p approaching 1 and infinity.

Cibulka Radek

Newton's method for solving generalized equations using setvalued approximations


Given Banach spaces $X$ and $Y$, a singlevalued mapping $f: X \to Y$ and a multivalued mapping $F:X\rightrightarrows Y$, we investigate the convergence properties of Newtontype iterative process for solving the generalized equation. The problem is to
\begin{equation}\label{Eqn1}
\mbox{find}\quad x\in X \quad \mbox{such that}\quad 0\in f(x)+F(x).
\end{equation}
This model has been used to describe
in a unified way various problems such as equations, inequalities, variational inequalities, and in particular,
optimality conditions. We study the following iterative process:
{\it Choose a sequence of setvalued mappings $A_k: X\times X\rightrightarrows Y$ approximating the function $f$ and a starting point $x_0 \in X$, and generate a sequence $(x_k)$ in $X$ iteratively by taking $x_{k+1}$ to be a solution to the auxiliary generalized equation
\begin{equation}\label{NewtonSeq}
0\in A_k(x_{k+1},x_k)+F(x_{k+1}) \quad \mbox{for each} \quad \quad k \in \{0,1,2, \dots\}.
\end{equation}}
In the first part, we present a result concerning the stability of metric regularity under setvalued perturbations. In the latter, this statement is applied in the study of (super)linear convergence of the iterative process (\ref{NewtonSeq}). Also some particular cases are discussed in detail.
This work is based on a forthcoming joint paper with Samir Adly and Huynh Van Ngai.

Dhara Raj Narayan

On one extension theorem dealing with weighted OrliczSlobodetskii space on the boundary of domain


Let $\Omega$ be a given domain with the sufficiently regular boundary. We
prove that every function belonging to a certain weighted OrliczSlobodetskii
space defined on the boundary of $\Omega$ can be extended to a function belonging to a
relevant weighted OrliczSobolev space defined on the whole $\Omega$. Analysis of the
admitted weights is provided and no restrictions on generating Orlicz function
is required.
References
[1] E. GAGLIARDO, Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di
funzioni in n variabili, (Italian) Rend. Sem. Mat. Univ. Padova 27 1957 284–305.
[2] J.P. GOSSEZ, Nonlinear elliptic boundary value problems for equations with rapidly (or
slowly) increasing coefficients, Trans. Amer. Math. Soc. 190 (1974), 163–205.
[3] A. KALAMAJSKA and M. KRBEC, Traces of OrliczSobolev functions under general growth
restrictions, Mathematishe Nachrichten 286(7) (2013), 730–742.
[4] M.TH. LACROIX, Espaces de traces des espaces de SobolevOrlicz, J. Math. Pures Appl.
53(9) (1974), 439–458.

Drábek Pavel

Necessary and sufficient condition for the discreteness of the spectrum of the pLaplacian with weights on radial domains

Heinz Sebastian

On a way to control oscillations for a special evolution equation


We study a rateindependent evolutionary system which models phase
transformations in elastoplasticity, as discussed in Mielke Theil
Levitas '02. We prove the existence of solutions. In order to do that,
we have to control the spatial oscillations of the phase indicator. Our
strategy combines two concepts: socalled mutual recovery sequences
(see, for example, Mielke Roubicek Stefanelli '08) and Tartar's
Hmeasures.

Hencl Stanislav

Optimal assumptions for discreteness


In the pioneering works J. Ball studied a class of mappings that could be
used to model nonlinear elasticity and he found weak conditions for
regularity and invertibility properties. One of the main properties in the
models of nonlinear elasticity is that there is no interpenetration of
matter. This in the physically relevant models corresponds to the fact
that two parts of the body cannot be mapped to the same place. From the
mathematical point of view this means that the map is onetoone and thus
invertible.
Let $\Omega\subset \mathbb{R^n}$ and $f:\Omega\to\mathbb{R^n}$ be a
mapping. In this talk we discuss the optimal assumptions that guarantee
that the mapping is open and discrete which is the main ingredient for
showing that it is invertible and in fact a homeomorphism. The optimal
condition in the plane for mappings of finite distortion was found by
Iwaniec and \v{S}ver\'{a}k in 1993 and they conjectured the optimal
assumption in higher dimension. Many people have contributed to the study
of this problem and recently we have found out that there is a
counterexample to dicreteness in dimension $n\geq 3$. This is a joint
result with Kai Rajala.

Izydorek Marek

Planar Newtonian systems with a singular potential; a geometric approach


We will be concerned with homoclinic and heteroclinic solutions of a second order planar Newtonian system with potential having a single well of infinite depth. Application of the Shadowing Chain Lemma to multiplicity results will be discussed.

Janczewska Joanna

Application of the CrandallRabinowitz theorem on simple bifurcation to nonlinear ploblems


The purpose of the lecture is to present a variational version
of the CrandallRabinowitz theorem on the existence of a simple bifurcation
point. We show how it may be applied to the study of bifurcation
in nonlinear problems of elasticity theory, for instance in von K\'{a}rm\'{a}n
equations.

Łukaszewicz Grzegorz

Attractors in problems of contact mechanics

Malý Jan

Sobolev maps with constraints


We establish a ``low rank property'' for Sobolev mappings that pointwise solve a
first order nonlinear system of PDEs, whose smooth solutions have the socalled
``contact property''. Namely, we consider a Sobolev mapping $f$
of an open set of $\mathbb R^m$, $n \less m \le 2n$, taking values in the Heisenberg group $\H^n$. If $f$ is horizontal,
then the Jacobi matrix of $f$ (regarded as $\mathbb R^{2n+1}$valued)
has rank at most $n$ a.e. (This result can be alternatively
explained without any reference to the Heisenberg group).
Contrarily, if $f$ has almost everywhere maximal rank,
its image must have a positive $(m+1)$dimensional Hausdorff measure with respect to the subRiemannian distance of $\H^n$.
This is a joint work with Valentino Magnani and Samuele Mongodi.

Mazowiecka Katarzyna

Generic singularities of harmonic and biharmonic maps


We will review the known results of HardtLin and AlmgrenLieb on singularities of harmonic maps
from the ball to the sphere, i.e.\;critical points of the Dirichlet integral
$E(u)=\int_{\mathbb{B}^3}\nabla u^2 dx$. \\
In the second part of the poster we will discuss the conjectures and work in progress on
generalization of this problems to biharmonic maps, namely critical points of functional
$\int_{\mathbb{B}^5}\Delta u^2 dx$ with nonlinear pointwise constraints $u(x)\in\mathbb{S}^4$.
\noindent
\large
\begin{thebibliography}{99}
\bibitem[1]{1} Almgren, F.,Lieb, E.,\textit{Singularities of energy minimizing maps from the ball to
the sphere: examples, counterexamples, and bounds}, Ann. of Math. (2), \textbf{128}, No. 3,
483530 (1988).
\bibitem[2]{2} Hardt, R., Lin FH. \textit{A remark on $H^1$ mappings}, Manuscripta Math.,
\textbf{56}, No. 1, 110 (1986).
\end{thebibliography}

Nayar Piotr

On a certain Sobolevtype inequality on L_1(T)


I will prove an upper bound for the norm of an operator IdA,
where A is a "good" convolution operator acting on the space of functions from L_1(T) with 0 mean.

Nečasová Šárka

Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains and incompressible limits


We consider the compressible (barotropic) NavierStokes system on timedependent domains,
supplemented with slip boundary conditions. Our approach is based on penalization of the
boundary behaviour, viscosity, and the pressure in the weak formulation. Globalintime weak
solutions are obtained. Secondly, we suppose that the characteristic speed of the fluid is dominated
by the speed of sound and perform the low Mach number limit in the framework of weak
solutions. The standard incompressible NavierStokes system is identified as the target problem.

Orpel Aleksandra

Boundary value problems with onedimensional phiLaplace operator

Osekowski Adam

Square function inequalities in BMO spaces


We will present a method for the simultaneous study of a BMO
function $\varphi$ and its dyadic square function $S(\varphi)$ that can yield sharp norm
inequalities between the two. The technique reduces the problem of proving an estimate to that of solving the corresponding nonlinear problem. One of the applications is the sharp bound for the
$p$th moment of $S(\varphi)$, $0 p \infty$, which in turn implies the squareexponential
integrability of the square function.

Peszek Jan

Interplay between flocking particles and a nonnewtonian viscous fluid


We will present an interplay between flocking particles and an incompressible viscous nonNewtonian fluid.
The flocking is described by the CuckerSmale's flocking model, which in our case is associated with a Vlasov type equation.
The fluid is described by a classic hydrodynamics' equation with the stress tensor represented by pLaplacian.
We will compare our results to the coupled VlasovNavierStokes system and CuckerSmaleNavierStokes system.

Pierzchalski Antoni

On conjugate sumbersions


We define and investigate pairs of $(p, q)$conjugate submersions of a Riemannian manifold, $\frac{1}{p}+\frac{1}{q}=1$,
and  in particular  of $(p, q)$conjugate functions. We show that conjugate submersions of the plane
are $p$ and $q$harmonic maps, respectively. Recall the that a map $\varphi:M\to \mathbb{R}$ is called to be $p$\textit{harmonic} if
$$
\mbox{div}(\nabla \varphi^{p2}\nabla\varphi) = 0
$$
where div is divergence operator. We show also that  in the case on
an arbitrary Riemannian manifold  the product of moduli of foliations
defined by conjugate submersions is equal to $1$.

Radice Teresa

The Maximum Principle of Alexandrov for very weak solutions


We extend the classical maximal principle of Alexandrov to the solutions of the elliptic equation
$\displaystyle{\sum_{i,j=1}^n} a_{ij}(x) \frac{\partial^2
u}{\partial x_i \partial x_j}=f$ whose second derivatives belong to the Zygmund space $\mathscr L^n \log^{\alpha}\mathscr L$, $\alpha=\frac{1}{n}$. The results obtained are in collaboration with Gabriella Zecca.

Ryabukha Tatiana

STABILITY OF SOLUTIONS OF KINETIC EQUATIONS CORRESPONDING TO THE REPLICATOR DYNAMICS


The replicator equation is a deterministic nonlinear equation arising in evolutionary game theory describing the evolving lifeforms in terms of frequencies of strategies. It is related to a mean field approach and therefore it has a macroscopic character: the description is referred to the frequencies (densities) of agents playing the corresponding strategies. However, the macroscopic approach is not sufficient to describe the dynamics of complex living systems by reducing the complexity of the overall systems. In some applications to consider the agents as discrete interacting units is important in order to capture the complexity of (biological) phenomena.
We propose a class of kinetic type equations that describe the replicator dynamics at the mesoscopic level. Under suitable assumptions we show the asymptotic (exponential) stability of the solutions to such kinetic equations in the case when the corresponding macroscopic equation is asymptotically stable. To obtain the mesoscopic model corresponding to the replicator equation we follow the techniques developed by N. Bellomo with coautors applying tools of the Kinetic Theory of active particles for complex living systems.
In perspective, the obtained results could be used for analysing the asymptotic behaviour in time of mathematical model which describes tumourimmune system competition.

Sasi Sarath  Semipositone Problems on Exterior Domains


We study nonnegative radial solutions to the problem
D u = L K(x) f(u); x in B'
u = 0 if x = r0
u > 0 as x > infinity
where L is a positive parameter, D is the Laplacian of u, B'={x in R^n: n>2, x>r0} and K belongs to a class of functions such that K(r) > 0 as r> infinity. For classes of nonlinearities f that are negative at the origin we discuss existence and uniqueness results.

Skrzypczak Iwona

Hardy inequalities dictated by nonlinear problems


We consider the anticoercive partial differential inequality of elliptic type involving $p$Laplacian: ${}\Delta_pu\ge \Phi$, where $\Phi$ is a given locally integrable function and $u$ is defined on an open subset $\Omega\subseteq\rn$. Knowing solutions, we derive Hardy inequalities involving certain measures for compactly supported Lipschitz functions. Our methods allow to retrieve classical Hardy inequalities with optimal constants and various weighted Hardy inequalities.

Slavíková Lenka

Sobolev algebras


We characterize when a general Sobolev space is a Banach algebra.

Stehlík Petr

Nonlinear difference equations  obstacles and their origin


Because of the finite dimension of functions spaces, problems in the theory of nonlinear difference equations are usually rather straightforward (compared to their continuous counterparts). However, there are many situations in which simple questions from the theory of differential equations are not properly answered in the discrete case. The goal of this talk is to present a couple of such problems (maximum principles and Fucik spectrum). We also discuss some recent tools which enable to properly analyze and identify similar phenomena (dynamic and generalized equations, etc.).

Veetil Anoop Thazhe

On Generalized HardySobolev Inequalities

Zappale Elvira

Relaxation for an optimal design problem with perimeter penalization


The results deal with relaxation and integral representation in the space of functions of
bounded variation, which find application in the framework of optimal design problems.
Our description takes into account a perimeter penalization term for the design region.
