Past Partial Differential Equations Seminar

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).

The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.

The description of the problem is indeed extremely complex in its generality.

In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.

Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.

The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast.  The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. 

The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential.    

Please note that this seminar has been cancelled due to unforeseen circumstances.

In this talk, we will present some recent mathematical features around two-fluid models. Such systems may be encountoured for instance to model internal waves, violent aerated flows, oil-and-gas mixtures. Depending on the context, the models used for simulation may greatly differ. However averaged models share the same structure. Here, we address the question whether available mathematical results in the case of a single fluid governed by the compressible barotropic equations for single flow may be extended to two phase model and discuss derivations of well-known multi-fluid models from single fluid systems by homogeneization (assuming for instance highly oscillating density). We focus on existence of local existence of strong solutions, loss of hyperbolicity, global existence of weak solutions, invariant regions, Young measure characterization.

Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion\emph{ Poiseuille Free Boundary} PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \par\noindentBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, however\emph{surface waves of finite wavelengths are observed}, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.
Aim of this talk is the investigation of nonlinear stability ofPFB providing \emph{ a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane}, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz \emph{nonlinear stability}. We use\emph{a modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability}. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\par\noindentNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. \\ {\bf References}\\ [1]  M.K. Smith, \textit{The mechanism for the long-waveinstability in thin liquid films} J. Fluid Mech., \textbf{217},1990, pp.469-485.
\\ [2]  Benjamin T.B., \textit{Wave formation in laminar flow down aninclined plane}, J. Fluid Mech. \textbf{2}, 1957, 554-574.
\\ [3]  Yih Chia-Shun, \textit{Stability of liquid flow down aninclined plane}, Phys. Fluids, \textbf{6}, 1963, pp.321-334.
\\ [4] Padula M., {\it On nonlinear stability of MHD equilibriumfigures}, Advances in Math. Fluid Mech., 2009, 301-331.
\\ [5] Padula M., \textit{On nonlinear stability of linear pinch},Appl. Anal.  90 (1), 2011, pp. 159-192.

We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution).

Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field.

This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma).

Fluids that are not adequately described by a linear constitutive relation are usually referred to as   "non-Newtonian fluids". In the last 15 years we have seen a significant progress in the mathematical theory of generalized Newtonian fluids, which is an important subclass of non-Newtonian fluids. We present some recent results in the existence theory and in the error analysis for approximate solutions. We will also indicate how these techniques can be generalized to more general constitutive relations.

The space BD of functions of bounded deformation consists of all L1-functions whose distributional symmetrized derivative (defined by duality with the symmetrized gradient ($\nabla u + \nabla u^T)/2$) is representable as a finite Radon measure. Such functions play an important role in a variety of variational models involving (linear) elasto-plasticity. In this talk, I will present the first general lower semicontinuity theorem for symmetric-quasiconvex integral functionals with linear growth on the whole space BD. In particular we allow for non-vanishing Cantor-parts in the symmetrized derivative, which correspond to fractal phenomena. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, based on local rigidity arguments for some differential inclusions involving symmetrized gradients. This strategy allows us to establish the lower semicontinuity result even without a BD-analogue of Alberti's Rank-One Theorem in BV, which is not available at present. A similar strategy also makes it possible to give a proof of the classical lower semicontinuity theorem in BV without invoking Alberti's Theorem.

In this talk we will first consider the Allen-Cahn action functional that controls the probability of rare events in an Allen-Cahn type equation with additive noise. Further we discuss a perturbation of the Allen-Cahn equation by a stochastic flow. Here we will present a tightness result in the sharp interface limit and discuss the relation to a version of stochastically perturbed mean curvature flow. (This is joint work with Luca Mugnai, Leipzig, and Hendrik Weber, Warwick.)

The classical isoperimetric inequality states that, given a set $E$ in $R^n$ having the same measure of the unit ball $B$, the perimeter $P(E)$ of $E$ is greater than the perimeter $P(B)$ of $B$. Moreover, when the isoperimetric deficit $D(E)=P(E)-P(B)$ equals 0, than $E$ coincides (up to a translation) with $B$. The sharp quantitative form of the isoperimetric inequality states that $D(E)$ can be bound from below by $A(E)^2$, where the Fraenkel asymmetry $A(E)$ of $E$ is defined as the minimum of the volume of the symmetric difference between $E$ and any translation of $B$. This result, conjectured by Hall in 1990, has been proven in its full generality by Fusco-Maggi-Pratelli (Ann. of Math. 2008) via symmetrization arguments and more recently by Figalli-Maggi-Pratelli (Invent. Math. 2010) through optimal transportation techniques. In this talk I will present a new proof of the sharp quantitative version of the isoperimetric inequality that I have recently obtained in collaboration with G.P.Leonardi (University of Modena e Reggio). The proof relies on a variational method in which a penalization technique is combined with the regularity theory for quasiminimizers of the perimeter. As a further application of this method I will present a positive answer to another conjecture posed by Hall in 1992 concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.

I will show that one can (at least in theory) guarantee the "validity" of a numerical approximation of a solution of the 3D Navier-Stokes equations using an explicit a posteriori test, despite the fact that the existence of a unique solution is not known for arbitrary initial data.

The argument relies on the fact that if a regular solution exists for some given initial condition, a regular solution also exists for nearby initial data ("robustness of regularity"); I will outline the proof of robustness of regularity for initial data in $H^{1/2}$.

I will also show how this can be used to prove that one can verify numerically (at least in theory) the following statement, for any fixed R > 0: every initial condition $u_0\in H^1$ with $\|u\|_{H^1}\le R$ gives rise to a solution of the unforced equation that remains regular for all $t\ge 0$.

This is based on joint work with Sergei Chernysehnko (Imperial), Peter Constantin (Chicago), Masoumeh Dashti (Warwick), Pedro Marín-Rubio (Seville), Witold Sadowski (Warsaw/Warwick), and Edriss Titi (UC Irivine/Weizmann).

In this talk we will present some recent results on the long time

asymptotics of the generalized (non-Markovian) Langevin equation (gLE). In particular,

we will discuss about the ergodic properties of the gLE and present estimates on the rate of convergence to equilibrium, we will present

a homogenization result (invariance principle) and we will discuss

about the convergence of the gLE dynamics to the (Markovian) Langevin

dynamics, in some appropriate asymptotic limit. The analysis is based on the approximation of the gLE by a

high (and possibly infinite) dimensional degenerate Markovian system,

and on the analysis of the spectrum of the generator of this Markov

process. This is joint work with M. Ottobre and K. Pravda-Starov.

A basic example of  shear flow wasintroduced  by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is  having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

We consider the Relativistic Vlasov-Maxwell system of equations which

describes the evolution of a collisionless plasma. We show that under

rather general conditions, one can test for linear instability by

checking the spectral properties of Schrodinger-type operators that

act only on the spatial variable, not the full phase space. This

extends previous results that show linear and nonlinear stability and

instability in more restrictive settings.

We will describe some joint work with V. G. Maz’ya and I. E. Verbitsky, concerning homogeneous quasilinear differential operators. The model operator under consideration is:

\[ L(u) = - \Delta_p u - \sigma |u|^{p-2} u. \]

Here $\Delta_p$ is the p-Laplacian operator and $\sigma$ is a signed measure, or more generally a distribution. We will discuss an approach to studying the operator L under only necessary conditions on $\sigma$, along with applications to the characterisation of certain Sobolev inequalities with indefinite weight. Many of the results discussed are new in the classical case p = 2, when the operator L reduces to the time independent Schrödinger operator.

We investigate ground state configurations of atomic pair potential systems in two dimensions as the number of particles tends to infinity. Assuming crystallization (which has been proved for some cases such as the Radin potential, and is believed to hold more generally), we show that after suitable rescaling, the ground states converge to a unique macroscopic Wulff shape. Moreover, we derive a scaling law for the size of microscopic non-uniqueness which indicates larger fluctuations about the Wulff shape than intuitively expected.

Joint work with Yuen Au-Yeung and Bernd Schmidt (TU Munich),

to appear in Calc. Var. PDE

We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett [Imperial College London].

It will be shown how the critical mass classical Keller-Segel system and

the critical displacement convex fast-diffusion equation in two

dimensions are related. On one hand, the critical fast diffusion

entropy functional helps to show global existence around equilibrium

states of the critical mass Keller-Segel system. On the other hand, the

critical fast diffusion flow allows to show functional inequalities such

as the Logarithmic HLS inequality in simple terms who is essential in the

behavior of the subcritical mass Keller-Segel system. HLS inequalities can

also be recovered in several dimensions using this procedure. It is

crucial the relation to the GNS inequalities obtained by DelPino and

Dolbeault. This talk corresponds to two works in preparation together

with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.

Let $u \in W^{1,p}(\Omega,\R^N)$, $\Omega$ a bounded domain in

$\R^n$, be a minimizer of a convex variational integral or a weak solution to

an elliptic system in divergence form. In the vectorial case, various

counterexamples to full regularity have been constructed in dimensions $n

\geq 3$, and it is well known that only a partial regularity result can be

expected, in the sense that the solution (or its gradient) is locally

continuous outside of a negligible set. In this talk, we shall investigate

the role of the space dimension $n$ on regularity: In arbitrary dimensions,

the best known result is partial regularity of the gradient $Du$ (and hence

for $u$) outside of a set of Lebesgue measure zero. Restricting ourselves to

the partial regularity of $u$ and to dimensions $n \leq p+2$, we explain why

the Hausdorff dimension of the singular set cannot exceed $n-p$. Finally, we

address the possible existence of singularities in two dimensions.

We study the nonhomogeneous boundary value problem for the

Navier--Stokes equations

\[

\left\{ \begin{array}{rcl}

-\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=&{0}\qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\bf u}&=&{\bf a} \qquad \hbox{\rm on }\;\;\partial\Omega

\end{array}\right

\eqno(1)

\]

in a bounded multiply connected domain

$\Omega\subset\mathbb{R}^n$ with the boundary $\partial\Omega$,

consisting of $N$ disjoint components $\Gamma_j$.

Starting from the famous J. Leray's paper published in 1933,

problem (1) was a subject of investigation in many papers. The

continuity equation in (1) implies the necessary solvability

condition

$$

\int\limits_{\partial\Omega}{\bf a}\cdot{\bf

n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf

n}\,dS=0,\eqno(2)

$$

where ${\bf n}$ is a unit vector of the outward (with respect to

$\Omega$) normal to $\partial\Omega$. However, for a long time

the existence of a weak solution ${\bf u}\in W^{1,2}(\Omega)$ to

problem (1) was proved only under the stronger condition

$$

{\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad

j=1,2,\ldots,N. \eqno(3)

$$

During the last 30 years many partial results concerning the

solvability of problem (1) under condition (2) were obtained. A

short overview of these results and the detailed study of problem

(1) in a two--dimensional bounded multiply connected domain

$\Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset

\Omega_1$ will be presented in the talk. It will be proved that

this problem has a solution, if the flux ${\cal

F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS$ of the

boundary datum through $\partial\Omega_2$ is nonnegative (outflow

condition).

The maps $u$ which are continuous in ${\mathbb R}^n$ and circle-valued are precisely the maps of the form $u=\exp (i\varphi)$, where the phase $\varphi$ is continuous and real-valued.

In the context of Sobolev spaces, this is not true anymore: a map $u$ in some Sobolev space $W^{s,p}$ need not have a phase in the same space. However, it is still possible to describe all the circle-valued Sobolev maps. The characterization relies on a factorization formula for Sobolev maps, involving three objects: good phases, bad phases, and topological singularities. This formula is the analog, in the circle-valued context, of Weierstrass' factorization theorem for holomorphic maps.

The purpose of the talk is to describe the factorization and to present a puzzling byproduct concerning sums of Dirac masses.

I will make a short review of some continous approximations to the Navier-Stokes equations, especially with the aim of introducing alpha models for the Large Eddy Simulation of turbulent flows.

Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations.

We develop a theory based on relative entropy to show stabilityand uniqueness of extremal entropic Rankine-Hugoniot discontinuities forsystems of conservation laws (typically 1-shocks, n-shocks, 1-contactdiscontinuities and n-contact discontinuities of big amplitude), amongbounded entropic weak solutions having an additional strong traceproperty. The existence of a convex entropy is needed. No BV estimateis needed on the weak solutions considered. The theory holds withoutsmallness condition. The assumptions are quite general. For instance, thestrict hyperbolicity is not needed globally. For fluid mechanics, thetheory handles solutions with vacuum.

The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).

Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.

In this talk we describe some recent work on shock

reflection problems for the potential flow equation. We will

start with discussion of shock reflection phenomena. Then we

will describe the results on existence, structure and

regularity of global solutions to regular shock reflection. The

approach is to reduce the shock reflection problem to a free

boundary problem for a nonlinear elliptic equation, with

ellipticity degenerate near a part of the boundary (the sonic

arc). We will discuss techniques to handle such free boundary

problems and degenerate elliptic equations. This talk is based

on joint works with Gui-Qiang Chen, and with Myoungjean Ba

• (a) nonconvergence for the unregularized mathematical problem or
• (b) nonuniqueness of the limit if it exists, or
• (c) limiting solutions only in the very weak form of a space time dependent probability distribution.

I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.

We investigate a dynamic model of two dimensional crystal growth

described by a forward-backward parabolic equation. The ill-posed

region of the equation describes the motion of corners on the surface.

We analyze a fourth order regularized version of this equation and

show that the dynamical behavior of the regularized corner can be

described by a traveling wave solution. The speed of the wave is found

by rigorous asymptotic analysis. The interaction between multiple

corners will also be presented together with numerical simulations.

This is joint work in progress with Fang Wan.

We consider the motion of a viscous incompressible fluid described by

the Navier-Stokes equations in a bounded cylinder with slip boundary

conditions. Assuming that $L_2$ norms of the derivative of the initial

velocity and the external force with respect to the variable along the

axis of the cylinder are sufficiently small we are able to prove long

time existence of regular solutions. By the regular solutions we mean

that velocity belongs to $W^{2,1}_2 (Dx(0,T))$ and gradient of pressure

to $L_2(Dx(0,T))$. To show global existence we prolong the local solution

with sufficiently large T step by step in time up to infinity. For this purpose

we need that $L_2(D)$ norms of the external force and derivative

of the external force in the direction along the axis of the cylinder

vanish with time exponentially.

Next we consider the inflow-outflow problem. We assume that the normal

component of velocity is nonvanishing on the parts of the boundary which

are perpendicular to the axis of the cylinder. We obtain the energy type

estimate by using the Hopf function. Next the existence of weak solutions is

proved.

Let $\phi$ be a convex, $C^1$-function and consider the functional: $$ (1)\qquad \mathcal{F}(\bf u)=\int_{\Omega} \phi (|\nabla \bf u|) \,dx $$ where $\Omega\subset \mathbb{R}^n$ is a bounded open set and $\bf u: \Omega \to \mathbb{R}^N$. The associated Euler Lagrange system is $$ -\mbox{div} (\phi' (|\nabla\bf u|)\frac{\nabla\bf u}{|\nabla\bf u|} )=0 $$ In a fundamental paper K.~Uhlenbeck proved everywhere $C^{1,\alpha}$-regularity for local minimizers of the $p$-growth functional with $p\ge 2$. Later on a large number of generalizations have been made. The case $1

{\bf Theorem.} Let $\bfu\in W^{1,\phi}_{\loc}(\Omega)$ be a local minimizer of (1), where $\phi$ satisfies suitable assumptions. Then $\bfV(\nabla \bfu)$ and $\nabla \bfu$ are locally $\alpha$-Hölder continuous for some $\alpha>0$.

We present a unified approach to the superquadratic and subquadratic $p$-growth, also considering more general functions than the powers. As an application, we prove Lipschitz regularity for local minimizers of asymptotically convex functionals in a $C^2$ sense.

Due to the scaling properties of the Navier-Stokes equations,

homogeneous initial data may lead to forward self-similar solutions.

When the initial data is small enough, it is well known that the

formalism of mild solutions (through the Picard-Duhamel formula) give

such self-similar solutions. We shall discuss the issue of large initial

data, where we can only prove the existence of weak solutions; those

solutions may lack self-similarity, due to the fact that we have no

results about uniqueness for such weak solutions. We study some tools

which may be useful to get a better understanding of those weak solutions.

In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion.

\textbf{Theorem 1}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain (open connected set) $\Omega\subset\mathbb{R}^2$. Suppose

$$ (1)\qquad \operatorname{Int} \nabla v(\Omega)=\emptyset. $$

Then $\operatorname{meas}\nabla v(\Omega)=0$.

Here $\operatorname{Int}E$ is the interior of ${E}$, $\operatorname{meas} E$ is the Lebesgue measure of ${E}$. Theorem 1 is a straight consequence of the following two results.

\textbf{ Theorem 2 [2]}.  Let $v:\Omega\to\mathbb{R}$ be a $C^1$-smooth function on a domain $\Omega\subset\mathbb{R}^2$. Suppose (1) is fulfilled. Then the graph of $v$ is a normal developing surface. 

Recall that a $C^1$-smooth manifold $S\subset\mathbb{R}^3$ is called  a normal developing surface [3] if for any $x_0\in S$ there exists a straight segment $I\subset S$ (the point $x_0$ is an interior point of $I$) such that the tangent plane to $S$ is stationary along $I$.

\textbf{Theorem 3}.  The spherical image of any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ has the area (the Lebesgue measure) zero.

Recall that the spherical image of a surface $S$ is the set $\{\mathbf{n}(x)\mid x\in S\}$, where $\mathbf{n}(x)$ is the unit normal vector to $S$ at the point~$x$. From Theorems 1--3 and the classical results of A.V. Pogorelov (see [4, Chapter 9]) we obtain the following corollaries Corollary 4. Let the spherical image of a $C^1$-smooth surface $S\subset\mathbb{R}^3$ have no interior points. Then this surface is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{ Corollary 5}. Any $C^1$-smooth normal developing surface $S\subset\mathbb{R}^3$ is a surface of zero extrinsic curvature in the sense of Pogorelov.

\textbf{Theorem 6}. Let $K\subset\mathbb{R}^{2\times 2}$ be a compact set and the topological dimension of $K$ equals 1. Suppose there exists $\lambda> 0$ such that $\forall A,B\in K, \, \, |A-B|^2\le\lambda\det(A-B).$

Then for any Lipschitz mapping $f:\Omega\to\mathbb R^2$ on a domain $\Omega\subset\mathbb R^2$ such that $\nabla f(x)\in K$ a.e. the identity $\nabla f\equiv\operatorname{const}$ holds.

Many partial cases of Theorem 6 (for instance, when $K=SO(2)$ or $K$ is a segment) are well-known (see, for example, [5]).

The author was supported by the Russian Foundation for Basic Research (project no. 08-01-00531-a).

[1] {Korobkov M.\,V.,} {``Properties of the $C^1$-smooth functions whose gradient range has topological dimension~1,'' Dokl. Math., to appear.}

[2] {Korobkov M.\,V.} {``Properties of the $C^1$-smooth functions with nowhere dense gradient range,'' Siberian Math. J., \textbf{48,} No.~6, 1019--1028 (2007).}

[3] { Shefel${}'$ S.\,Z.,} {``$C^1$-Smooth isometric imbeddings,'' Siberian Math. J., \textbf{15,} No.~6, 972--987 (1974).}

[4] {Pogorelov A.\,V.,} {Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs. Vol. 35. Providence, R.I.: American Mathematical Society (AMS). VI (1973).}

[5] {M\"uller ~S.,} {Variational Models for Microstructure and Phase Transitions. Max-Planck-Institute for Mathematics in the Sciences. Leipzig (1998) (Lecture Notes, No.~2. http://www.mis.mpg.de/jump/publications.html).}

In this talk I will present recent developments of the obstacle type problems, with various applications ranging

from: Industry to Finance, local to nonlocal operators, and one to multi-phases.

The theory has evolved from a single equation

$$

\Delta u = \chi_{u > 0}, \qquad u \geq 0

$$

to embrace a more general (two-phase) form

$$

\Delta u = \lambda_+ \chi_{u>0} - \lambda_- \chi_{u0$.

The above problem changes drastically if one allows $\lambda_\pm$ to have the incorrect sign (that appears in composite membrane problem)!

In part of my talk I will focus on the simple {\it unstable} case

$$

\Delta u = - \chi_{u>0}

$$

and present very recent results (Andersson, Sh., Weiss) that classifies the set of singular points ($\{u=\nabla u =0\}$) for the above problem.

The techniques developed recently by our team also shows an unorthodox approach to such problems, as the classical technique fails.

At the end of my talk I will explain the technique in a heuristic way.

Several dissipative scalar conservation laws share the properties of

$L1$-contraction and maximum principle. Stability issues are naturally

posed in terms of the $L1$-distance. It turns out that constants and

travelling waves are asymptotically stable under zero-mass initial

disturbances. For this to happen, we do not need any assumption

(smallness of the TW, regularity/smallness of the disturbance, tail

asymptotics, non characteristicity, ...) The counterpart is the lack of

a decay rate.

We consider monochromatic planar electro-magnetic waves propagating through a nonlinear dielectric medium in the optical regime.

Travelling waves are particularly simple solutions of this kind. Results on the existence of guided travelling waves will be reviewed. In the case of TE-modes, their stability will be discussed within the context of the paraxial approximation.

The talk starts with the observation that many well-known systems of diffusive type

can be written as Wasserstein gradient flows. The aim of the talk is

to understand _why_ this is the case. We give an answer that uses a

connection between diffusive PDE systems and systems of Brownian

particles, and we show how the Wasserstein metric arises in this

context. This is joint work with Johannes Zimmer, Nicolas Dirr, and Stefan Adams.

Certain elastic structures and growing tissues (leaves, flowers or marine invertebrates) exhibit residual strain at free equilibria. We intend to study this phenomena through an elastic growth variational model. We will first discuss this model from a differential geometric point of view: the growth seems to change the intrinsic metric of the tissue to a new target non-flat metric. The non-vanishing curvature is the cause of the non-zero stress at equilibria.

We further discuss the scaling laws and $\Gamma$-limits of the introduced 3d functional on thin plates in the limit of vanishing thickness. Among others, given special forms of growth tensors, we rigorously derive the non-Euclidean versions of Kirchhoff and von Karman models for elastic non-Euclidean plates. Sobolev spaces of isometries and infinitesimal isometries of 2d Riemannian manifolds appear as the natural space of admissible mappings in this context. In particular, as a side result, we obtain an equivalent condition for existence of a $W^{2,2}$ isometric immersion of a given $2$d metric on a bounded domain into $\mathbb R3$.

We investigate the minimization problem for the variational integral

$$\int_\Omega\sqrt{1+|Dw|^2}\,dx$$

in Dirichlet classes of vector-valued functions $w$. It is well known that

the existence of minimizers can be established if the problem is formulated

in a generalized way in the space of functions of bounded variation. In

this talk we will discuss a uniqueness theorem for these generalized

minimizers. Actually, the theorem holds for a larger class of variational

integrals with linear growth and was obtained in collaboration with Lisa

Beck (SNS Pisa).