Differential equations in theory and applications

The topic is differential equations (both partial and ordinary differential equations).

Fall term 2010

Unless otherwise noted, talks are Tuesdays, 14:15–15:00 in room 1329, S-2.
Talks are listed below in reverse chronological order:

Date Speaker and title
2010-11-30 Eugenia Malinnikova
Carleman estimates and quantitative unique continuation
Abstract: In the talk we give a short survey of the unique continuation for second order elliptic operators and discuss quantitative estimates for unique continuation. We present a scheme for quantitative unique continuation for operators with non-constant coefficients and singular lower order terms. Our starting point is the Carleman type estimate of Koch and Tataru. The three spheres theorem for solutions of the corresponding equations and the reverse Holder inequality are derived from the estimate. Then some ideas of Nadirashvili give a quantitative propagation of smallness from sets of positive Lebesgue measure. The talk is based on a joint work with S.Vessella.
2010-11-16 Hermano Frid
Homogenization of strongly degenerate porous medium type equations in ergodic algebras
Abstract: We present some recent results about homogenization of strongly degenerate porous medium type equations in ergodic algebras obtained in collaboration with J.C. Silva. In particular, we recall the concepts of algebras with mean value and ergodic algebras and show their relation with continuous stationary processes in compact spaces, and the equivalence between individual homogenization in ergodic algebras and stochastic homogenization of those processes.
2010-11-02 Helge Holden
On the nonlinear variational wave equation
Abstract: We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation \[u_{tt}-c(u)(c(u)u_x)_x=0.\] This is joint work with X. Raynaud.
2010-10-19 Erik Lindgren
A non-local and non-linear operator of infinite laplacian-type
Abstract: We consider the limit of minimizers of the following \(W^{s,p}\)-type seminorm \[ \iint_{\Omega^2}\frac{|u(x)-u(y)|^p}{|x-y|^{\alpha p}} d x d y, \] as \(p \to \infty\). For the case of the space \(W^{1,p}\), it is well known that, as \(p\to \infty\), the minimizers converge to an infinite harmonic function, which also minimizes the Lipschitz norm. In our case, it turns out that the minimizers converge to the solution of the equation \[ Lu=\sup_{y\in \Omega}\frac{u(x)-u(y)}{|x-y|^\alpha}+\inf_{y\in \Omega}\frac{u(x)-u(y)}{|x-y|^\alpha}=0, \] and also that the limit is a minimizer of the Hölder \(C^\alpha\)-seminorm.
We are also able to construct solutions for the inhomogeneous equation \[Lu=f\] whenever \(f \in C(\Omega)\cap L^\infty(\Omega)\) and obtain some regularity result of subsolutions and solutions.
2010-10-05 Vesa Julin
Convexity criteria and uniqueness of absolutely minimizing functions
Abstract: This is a joint work with Armstrong, Crandall and Smart. We show that absolutely minimizers of the functional \[\sup_{x \in U} H(Du(x))\] relative to a convex Hamiltonian H are uniquely determined by their boundary values under minimal assumptions on H. Along the way, we extend the known results between absolute minimizers and the convexity of the corresponding Hamilton–Jacobi flow.
2010-09-21 Peter Lindqvist
A curious equation
Abstract: The optimization problem \[\min_{u} \max_{x}\bigl(|\nabla u(x)|^{p(x)}\bigr)\] has an interesting Euler–Lagrange equation. We prove the uniqueness of its viscosity solutions.
2010-09-07 Simone Cifani
Fourier spectral approximation for periodic solutions of fractional integro-PDEs
Abstract: In this talk I will introduce a Fourier spectral approximation for periodic solutions of a multidimensional scalar conservation law appended with a generalized diffusion operator which is the generator of pure jump Levy processes. These equations appear in the study of anomalous convection-diffusion phenomena and reduce to the so-called fractal/fractional conservation laws whenever the diffusion operator is chosen as the fractional Laplacian. Solutions of such equations can develop shock discontinuities, thus making non-trivial every attempt to devise a spectrally accurate numerical approximation. First, I will introduce an entropy formulation for such equations which is well-suited for periodic solutions, and prove uniqueness. Next, to prove existence I will introduce a Fourier spectral vanishing viscosity approximation à la Tadmor. Such an approximation retains the spectral accuracy of the underlying Fourier approximation, but at the same time is also able to enforce the correct amount of entropy dissipation (which is otherwise missing in the standard Fourier method). Finally, I will present some numerical results on the so-called fractional Burgers' equation.
2010-11-02, Espen Robstad Jakobsen