Day 1 Titles and Abstracts



On some optimal partition problems

Susanna Terracini
University of Milano, Italy
susanna.terracini -at- gmail.com
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We consider the free boundary problem associated with linear and nonlinear eigenvalues. We are concerned with extremality conditions and the regularity of the nodal sets, also in connection with that of eigenfunctions and the number of their nodal domains.

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Existence of solutions to two phase free boundary problems and applications

Jyotshana Prajapat
The Petroleum Institute, Abu Dhabi
jprajapat -at- pi.ac.ae

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I prove existence of solutions for a variational functional related to two-phase free boundary problem. As an application, we obtain solutions for two-phase Quadrature domains.

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Functional-anatytic theory of concentration compactness and the Trudinger-Moser inequality

Kyril Tintarev
Uppsala University, Sweden
tintarev -at- math.uu.se

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An imbedding of two Banach spaces $ X\subset Y$ is called cocompact relatively to a group $ D$ of linear isometries on both $ X$ and $ Y$ if $ g_ku_k\rightharpoonup 0$ in $ X$ implies $ u_k\to 0$ in $ Y$. Many known imbeddings that are not compact, are cocompact, usually with respect to translations or dilations, in iparticular, Sobolev imbeddings (including subelliptic and fractional Sobolev spaces), imbeddings of Besov spaces, and Stricharz imbeddings. In this talk we discuss profile decomposition - a refinement of the Banach-Alaoglu theorem in presence of a cocompact imbedding and some of its applications to PDE. We discuss transformations on a unit disk involved in cocompact imbeddings of $ H_0^1(B)$ and show how requirement of invariance under these transformations yields an unconditional improvement of the Trudinger-Moser inequality.

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Higher integrability and higher differentiability for solutions to variational problems

Arrigo Cellina
University of Milan, Italy
arrigo.cellina -at- unimib.it

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We consider the properties of a solution $ u$ to the problem of minimizing

$\displaystyle \int_{\Omega} f (x , v (x), \Delta v (x)) dx$

on $ u_0 + W^{1,1} (\Omega),$ where $ f(x, u , \xi)$ is a function, convex in hte variable $ \xi$ and satisfying further regularity and growth properties. We are interested in particular, to the case where the growth of $ f$ in the variable $ \xi$ is faster thatn polynomial growth. We consider the question of the $ further integrability$ of $ u$ i.e., besides saying that $ f(\cdot u (\cdot), \nabla u (\cdot))$ is integrable, can one say that $ \vert \nabla f (\cdot , u (\cdot), \nabla u (\cdot) ) \Vert \nabla u (\cdot) \vert$ is integrable? And the question of the $ further differentiability$ of $ u$, i.e., can one say that $ \nabla u (\cdot)$ is in a Sobolev space?

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A stationary Cahn-Hilliard model with the $ p$-Laplacian:radial solutions and pattern formation

Peter Takac
Universität Rostock,Germany
peter.takac -at- uni-rostock.de
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We begin by establishing a sharp (optimal) $ W^{2,2}_{\mathrm{loc}}$-regularity result for bounded weak solutions to a nonlinear elliptic equation with the $ p$-Laplacian, $ \Delta_p u\stackrel{{\mathrm {def}}}{=}\mathop{\rm {div}}(\vert\nabla u\vert^{p-2}\nabla u)$, $ 1 < p< \infty$. We develop very precise, optimal regularity estimates on the ellipticity of this singular (for $ 1 < p < 2$) or degenerate (for $ 2 < p < \infty$) problem. We apply this regularity result to prove Pohozhaev's identity for the elliptic Neumann problem

$\displaystyle \nonumber - \Delta_p u + W'(u) = f(x) \;$ in $\displaystyle \Omega \,;\qquad {\partial u} / {\partial\nu} = 0 \;$ on $\displaystyle \partial\Omega \,.$ (P)

Here, $ \Omega$ is a bounded domain in $ \rm I\hskip-1.8pt R^N$ whose boundary $ \partial\Omega$ is a $ C^2$-manifold, $ \nu\equiv \nu(x_0)$ denotes the outer unit normal to $ \partial\Omega$ at $ x_0\in \partial\Omega$, $ x = (x_1,\dots,x_N)$ is a generic point in $ \Omega$, $ f\in L^{\infty}(\Omega)\cap W^{1,1}(\Omega)$, and $ u\in W_0^{1,p}(\Omega)$ is an unknown function. The potential $ W:\,\rm I\hskip-1.8pt R\to \rm I\hskip-1.8pt R$ is assumed to be of class $ C^1$ and of the typical double-well shape of type $ W(s) = \left\vert 1 - \vert s\vert^{\beta}\right\vert^{\alpha}$ for $ s\in \rm I\hskip-1.8pt R$, where $ \alpha, \beta > 1$ are some constants. Finally, we take an advantage of the Pohozhaev identity to show that Problem (P) with $ f\equiv 0$ in $ \Omega$ has no phase transition solution $ u\in W_0^{1,p}(\Omega)$ ( $ 1 < p\leq N$), such that $ -1\leq u\leq 1$ in $ \Omega$, $ u\equiv -1$ in $ \Omega_{-1}$, and $ u\equiv 1$ in $ \Omega_1$, where both $ \Omega_{-1}$ and $ \Omega_1$ are some nonempty subdomains of $ \Omega$. Such a scenario for $ u$ is possible only if $ N=1$ and $ \Omega_{-1}, \Omega_1$ are finite unions of suitable subintervals of the interval $ \Omega\subset \rm I\hskip-1.8pt R^1$.

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On 3-TGFES for a singularly perturbed transient convection-diffusion equation3C

Vivek Sangwan and B.V.Rathish Kumar
Indian Institute of Technology Kanpur, India
bvrk -at- iitk.ac.in
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in the present work, we propose a Three-Step Taylor Galerkin Finite element Scheme (3-TGFES) for a singularly Perturbed Convection-Diffusion Problem. Traditional finite element methods with linear shape functions do not give rise to uniformly convergent schemes for singularly perturbed differential equations on a uniform mesh. Here, we have used exponentially fitted shape functions to generate a uniformly convergent scheme. In Three-Step Taylor Galerkin Method time dicretization is carried out prior to spatial descretization. This leads to higher order accuracy in the numerical solution. Further, the method is also known for its inherent upwinding capability. In the present work, apriori-error estimates for the proposed scheme have been derived and it has been shown that the proposed method is of third order accurate in time and linear in space. numerical results are presented for a couple of convection-dominated singularly perturbed problems.

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Numerical shape optimization for compressible flows

C. Praveen
Tata Institute of fundamental Research
Centre for Applicable Mathematics, Bangalore, India
praveen -at- math.tifrbng.res.in
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Numerical approximation of PDE models in fluid flows can be used to optimize the flow for some given objectives. The control can be shape of the boundary or some other parameter. While adjoint based methods are being developed, they are still not widely in use due to some remaining issues. On the other hand, gradient-free methods are relatively easy to use for optimization. The talk will discuss these methods with some applications to aerodynamic problems.

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