## Wolfgang Pauli Institute (WPI) Vienna

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## Nonlinear Analysis (2005)

Organizers: Henrik Shahgholian (Stockholm), Peter Markowich (WPI c/o Univ. Vienna)

### Talks

 Apushkinskaya Darya WPI seminar room Thu, 21. Apr 05, 13:00 Steady states of anisotropic generalized Newtonian fluids We consider the stationary flow of a generalized Newtonian fluid which is modelled by an anisotropic dissipative potential $f$. More precisely, we are looking for a solution $u:\\Omega \\rightarrow \\mathbb{R}^n$, $\\Omega \\subset \\mathbb{R}^n$, $n=2,3$, of the following system of nonlinear partial differential equations \\left\\{\\begin{aligned}-\\text{div} \\{T(\\varepsilon (u))\\}+u^k\\frac{\\partial u}{\\partial x_k}+\\nabla \\pi &=g \\quad \\text{in}\\ \\Omega ,\\\\ \\text{div}\\ u=0 \\quad \\text{in}\\ \\Omega, \\qquad u&=0 \\quad \\text{on}\\\\partial \\Omega.\\end{aligned}\\right. \\eqno{(*)} Here $\\pi :\\Omega \\rightarrow \\mathbb{R}$ denotes the pressure, $g$ is a system of volume forces, and the tensor $T$ is the gradient of the potential $f$. Our main hypothesis imposed on $f$ is the existence of exponents $10$. Under natural assumptions on $p$ and $q_0$ we prove the existence of a weak solution $u$ to the problem (*), moreover we prove interior $C^{1,\\alpha }$-regularity of $u$ in the two-dimensional case. If $n=3$, then interior partial regularity is established. Thematic program: Nonlinear Analysis (2005)

 Bucur Dorin WPI seminar room Thu, 21. Apr 05, 14:00 A variational approach of the isoperimetric inequality for the buckling problem We discuss a possible issue to solve the isoperimetric problem for the buckling load of a clamped plate. This is to prove that the ball is solution of the following $$\\min _{\\Omega \\in \\R^2, |\\Omega| =c} \\min _{u \\in H^2_0 (\\Omega), u \\not= 0} \\frac{\\int_\\Omega |\\Delta u|^2 dx}{\\int_\\Omega |\\nabla u|^2 dx}.$$ In 1995 Willms and Weinberger proved that if this problem has a solution, which is a simply connected smooth set, then this solution is the ball. In this joint work with Mark Ashbaugh, we prove the existence of a solution and discuss the minimal necessary regularity in order to solve the conjecture. Thematic program: Nonlinear Analysis (2005)

 Matevosyan Norayr WPI seminar room Thu, 21. Apr 05, 15:00 Global solutions of an Obstacle-Problem like equation We will consider the following minimization problem: For given domain $D\\subset R^n$, minimize the functional $$J(v)=\\int_{D} (\\frac{|\\nabla v|^{2}}{2}+v^{+})$$ over the space $K=\\{v\\in W^{1,2}(D): v-u_0 \\in W^{1,2}_0(D)\\}$ for a given $u_0 \\in W^{1,2}(D)$. It is important to note that $u_0$ can take negative values. We give a review of the problem and a classification of the global solutions. Thematic program: Nonlinear Analysis (2005)

 Brenier, Yann (CNRS Nice) Seminar Room C 207 Tue, 19. Jul 05, 14:10 Blow up techniques for hyperbolic problems with kinetic formulations Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Manfredi, Juan (U Pittsburgh) Seminar Room C 207 Tue, 19. Jul 05, 16:30 The p-Laplacian Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Rascle, Michel (U. Nice) Seminar Room C 207 Wed, 20. Jul 05, 11:00 PDEs in Traffic Modeling Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Westdickenberg, Michael (MPI Bonn) Seminar Room C 207 Wed, 20. Jul 05, 14:30 Structure result for scalar conservation laws Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Bardos, Claude (Paris 7 and Lab. J.L. Lions) Seminar Room C 207 Thu, 21. Jul 05, 9:30 Analytic stability and singularities for Kelvin Helmholtz, Rayleigh Taylor, Problems - Comparison with the stability of water waves problems Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Vasseur, Alexis (UT Austin) Seminar Room C207 Thu, 21. Jul 05, 11:00 Strong traces for multivariable scalar conservation laws Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Dhamo, Elidon (U. Münster) Seminar Room C207 Mon, 25. Jul 05, 11:00 Dispersive effects in quantum kinetic equations Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Rascle, Michel (U. Nice) Seminar Room C207 Mon, 25. Jul 05, 14:30 Strong traces for scalar conservation laws Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Kamotski, Vladimir (Bath) Seminar Room C207 Mon, 25. Jul 05, 15:45 Some problems in homogenisation Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Stimming, Hans Peter (WPI & U. Wien) Seminar Room C207 Tue, 26. Jul 05, 14:30 Monotonicity Properties of Blowup time for Nonlinear Schroedinger Equations Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Fellner, Klemens (WPI & U. Wien) Seminar Room C207 Tue, 26. Jul 05, 15:45 Entropy methods for reaction diffusion equations Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Trabelsi, Saber (U.Wien & U. Paris 7) Seminar Room C207 Wed, 27. Jul 05, 11:00 MCTDHF : The Multi Configuration Time Dependent Hartree Fock Equations Note:   WPI Summer School + Workshop 2005 Thematic program: Nonlinear Analysis (2005)

 Caffarelli Luis (University of Texas at Austin) Nordbergstraße 15, Room C 207 Mon, 24. Oct 05, 10:00 Problems with constraint for the fractional Laplacian Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Souganidis Takis (University of Texas at Austin) Nordbergstrasse 15, Room C 207 Mon, 24. Oct 05, 11:00 Convergence and error estimates for approximations of viscosity solutions Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Shahgholian Henrik (KTS, Stockholm) Nordbergstrasse 15, Room C 207 Mon, 24. Oct 05, 15:00 Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Athanasopoulos Ioannis (University of Crete, Heraklion) Nordbergstrasse 16, Room C 206 Tue, 25. Oct 05, 9:30 Stefan-like problems Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Vazquez Juan Luis (Universidad Autonoma de Madrid) Nordbergstrasse, Room C 206 Tue, 25. Oct 05, 11:00 The log-diffusion equation: geometry and analysis Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Farina Alberto (CNRS/WPI) Nordbergstrasse, Room C 206 Tue, 25. Oct 05, 13:00 Some results concerning the Lame-Emden Equation Note:   Conference on nonlinear PDEs in honor of Luis Caffarelli Thematic program: Nonlinear Analysis (2005)

 Uraltseva, Nina WPI Seminarroom, Nordbergstrasse 15 Wed, 23. Nov 05, 11:00 Estimates for solutions to free boundary close to fixed boundary Thematic program: Nonlinear Analysis (2005) WK seminar

 Apushinskaya, Darya WPI Seminarroom, Nordbergstrasse 15 Wed, 23. Nov 05, 12:00 Estimates for solutions of quasilinear elliptic Dirichlet problems in convex domains Thematic program: Nonlinear Analysis (2005)

 Gustafsson Björn (KTH, Stockholm) Nordbergstrasse 15 , C714 Seminar room Tue, 13. Dec 05, 9:00 On the geometry of Laplacian growth I will discuss questions concerning the geometry of Laplacian growth, including some recent ideas due to Krichever, Marshakov, Mineev-Weinstein, Wiegmann, Zabrodin and others on more general Laplacian deformations. Thematic program: Nonlinear Analysis (2005)

 Vasiliev Alexander (University of Bergen) Nordbergstrasse 15, Room C 207 Tue, 13. Dec 05, 10:15 Lagrangian characteristics of simply connected Laplacian growth We discuss and construct an action functional (logarithmic action)for simply connected Laplacian growth and obtain its variation. This variation admits various interpretations. In particular, it represents the infinitesimal version of the action of the Virasoro-Bott group over the space of analytic univalent functions. Thematic program: Nonlinear Analysis (2005)

 Hedenmalm Håkan(KTH, Stockholm) Nordbergstrasse 15, Room C 207 Tue, 13. Dec 05, 11:45 Quantum Laplacian growth with extremal potential Thematic program: Nonlinear Analysis (2005)

 Kawohl, Bernd WPI Seminarroom C 714 Mon, 23. Jan 06, 14:00 Overdetermined boundary value problems and the p-Laplacian Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Brock, Friedemann WPI Seminarroom C 714 Mon, 23. Jan 06, 15:00 Asymptotic estimates at infinity for p-Laplace equations and symmetry for entire solutions Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Pacella, Filomena WPI Seminarroom C 714 Tue, 24. Jan 06, 9:30 Morse index and geometrical properties of solutions of semilinear elliptic equations Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Nyström, Kaj WPI Seminarroom C 714 Tue, 24. Jan 06, 11:00 On a parabolic symmetry problem Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Damascelli, Lucio WPI Seminarroom C 714 Tue, 24. Jan 06, 14:00 Regularity and qualitative properties of positive solutions of m-Laplace problems via weighted Sobolev spaces Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Sirakov, Boyan WPI Seminarroom C 714 Tue, 24. Jan 06, 15:00 Positivity and symmetry of viscosity solutions of Hamilton-Jacobi-Bellman equations Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Bandle, Catherine WPI Seminarroom C 714 Wed, 25. Jan 06, 9:00 Nonradial clustered spike solutions for semilinear elliptic problems on S^n Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Farina, Alberto WPI Seminarroom C 714 Wed, 25. Jan 06, 10:30 One-dimensional symmetry for solutions of quasilinear equations in the plane Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Henrot, Antoine WPI Seminarroom C 714 Wed, 25. Jan 06, 11:30 Symmetry problems related to eigenvalues Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Willem, Michel WPI Seminarroom C 714 Thu, 26. Jan 06, 9:30 Symmetry properties of solutions of semilinear elliptic problems Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Weth, Tobias WPI Seminarroom C 714 Thu, 26. Jan 06, 10:30 The shape of extremal functions in Poincaré-Sobolev-type inequalities Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 McKenna, Joe WPI Seminarroom C 714 Thu, 26. Jan 06, 14:00 Approximate symmetry for finite difference equations Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Sciunzi, Berardino WPI Seminarroom C 714 Thu, 26. Jan 06, 15:30 Flat level set regularity of p-Laplace phase transitions Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)

 Reichel, Wolfgang WPI Seminarroom C 714 Thu, 26. Jan 06, 16:30 Electrostatic characterization of the sphere Note:   Workshop "Symmetries of solutions of nonlinear elliptic PDEs" Thematic program: Nonlinear Analysis (2005)