Forthcoming Seminars › Seminar Series

Partial Differential Equations Seminar

Mon, 18/01/2010
17:00 		Henrik Shahgholian (KTH Stockholm) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Obstacle type problems : An overview and some recent results
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
$$ \Delta u = \lambda_+ \chi_{u>0} - \lambda_- \chi_{u<0} $$
with $ \lambda_\pm $ reasonably smooth functions (down to Dini continuous). Astonishing results of Yuval Peres and his collaborators has shown remarkable relationships between obstacle problem and various forms of random walks, including Smash sum of Diaconis-Fulton (Lattice sets), and there is more to come. The two-phase form (and its multi-phase form) has been under investigation in the last 10 years, and interesting recoveries has been made about the behavior of the free boundaries in such problems. Existing methods has so far only allowed us to consider $ \lambda_\pm>0 $. 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 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.
Mon, 25/01/2010
17:00 		Mikhail Korobkov (Sobolev Institute of Mathematics) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Properties of the $ C^1 $-smooth functions whose gradient range has topological dimension 1
In the talk we discuss some results of [1]. We apply our previous methods [2] to geometry and to the mappings with bounded distortion. 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. 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 $. 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. 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. 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 $-smooth functions with nowhere dense gradient range,” Siberian Math. J., 48, No.~6, 1019–1028 (2007).} [3] { Shefel$ {}' $ S.\,Z.,} {“$ C^1 $-Smooth isometric imbeddings,” Siberian Math. J., 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] {Müller ~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).}
Mon, 01/02/2010
17:00 		Pierre-Gilles Lemarié-Rieusset (Université d'Évry) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Large homogeneous initial data for the 3D Navier-Stokes equations
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.
Mon, 15/02/2010
17:00 		Bianca Stroffolini (University of Naples) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Regularity results for functionals with general growth
In this talk I will present some results on functionals with general growth, obtained in collaboration with L. Diening and A. Verde. 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.
Mon, 01/03/2010
17:00 		Marius Beceanu (l'École des Hautes Études en Sciences Sociales (EHESS)) Partial Differential Equations Seminar Add to calendar L2
Stability of solitons for the Schroedinger Equation in Three Dimensions
Mon, 08/03/2010
17:00 		Wojciech ZAJACZKOWSKI (Polish Academy of Sciences) Partial Differential Equations Seminar Add to calendar Gibson 1st Floor SR
Global regular solutions to the Navier-Stokes equations in a cylinder with slip boundary conditions
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.

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