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Title: Polarizing Carnot Groups Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
We explore ways to polarize Carnot groups.
Title: Randomness in the Heisenberg group Speaker: Diego Ricciotti Time: 12:30pm‐1:30pm Place: CMC 109
We discuss some stochastic processes and their relation to Brownian motion and PDEs in the Heisenberg group.
Title: A new proof for the equivalence of weak and viscosity solutions for the \(p\)-Laplace and the \(p(x)\)-Laplace equation in \(R^n\) Speaker: Robert Freeman Time: 12:30pm‐1:30pm Place: CMC 109
The equivalence of weak and viscosity solutions in \(R^n\) was given by Juutinen, Lindqvist, and Manfredi in 2001. In this talk we will examine the new proof of the equivalence for the \(p\)-Laplace equation in \(R^n\) via infimal convolutions given by Julin and Juutinen in 2012. We will also discuss the strategies/difficulties in extending this proof to the \(p(x)\)-Laplace equation in R^n as well as the Heisenberg group.
Title: Tug of war games and the p-Laplacian in the Heisenberg group Speaker: Diego Ricciotti Time: 12:30pm‐1:30pm Place: CMC 109
We discuss a game theoretic interpretation of \(p\)-harmonic functions by exploiting the connection between PDEs, mean value properties and dynamic programming principles. In particular we obtain an approximation scheme at a discrete level, and we prove uniform convergence to the solution of the associated Dirichlet problem with continuous boundary datum, under suitable regularity assumptions of the domain.
Title: Equivalence of weak and viscosity solutions to the \(p(x)\)-Laplace equation in Carnot groups Speaker: Robert Freeman Time: 12:30pm‐1:30pm Place: CMC 109
We will discuss the known Euclidean results and new results in Carnot groups of the equivalence of weak and viscosity solutions to the \(p(x)\)-Laplace equation.
No seminar this week due to Spring Break.
Title: Nonlinear potential theory, Part 1 Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
We discuss various topics in nonlinear potential theory from Heinonen-Kilpelainen-Martio such as \(A_p\) weights, quasiconformal mappings and capacity of sets.
Title: Plates with incompatible prestrain and non-euclidean elasticity Speaker: Diego Ricciotti Time: 12:30pm‐1:30pm Place: CMC 109
We examine a model (coming from non-Euclidean elasticity) for thin plates that become internally prestrained in the absence of external forces or imposed boundary conditions. Examples of this phenomena occur in nature for instance in growing leaves or tissues. We describe a derivation of a 2D model arising as \(\Gamma\)-limit of a 3D model as the thickness parameter approaches zero, under appropriate scalings of the energy in terms of the plate's thickness.
Title: Sub-Riemannian Geometry and Partial Differential Equations, Part IV Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
Title: Sub-Riemannian Geometry and Partial Differential Equations, Part III Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
Title: Sub-Riemannian Geometry and Partial Differential Equations, Part II Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
Title: Sub-Riemannian Geometry and Partial Differential Equations Speaker: Thomas Bieske Time: 12:30pm‐1:30pm Place: CMC 109
We examine properties of solutions to the eikonal and infinite Laplace equations in sub-Riemannian spaces. The interplay between the underlying geometry and these solutions will also be explored.