arXiv subjects
Publications and source records attributed to Jan Cristina.
We compare the regularity of the boundary of a convex set with the value of its Finslerian volume entropy. The main result states that the volume entropy of a two-dimensional domain whose associated curvature measure is Ahlfors $α$-regular is $\frac{2α}{α+1}$.
We study the evolution equation $\partial_{t}u=-Λ_{t}u$ where $Λ_ {t}$ is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary $Σ_{t}$. We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on $\mathcal{M}=Σ_{0}$ to the boundaries of $\partialΣ_{t}$. Consequently we are able to derive a lower bound for the difference of the Dirichlet-Neumann maps in terms of the difference of a background metrics $g$ and an inclusion metric $g+χ_Σ(h-g)$ on a manifold $\mathcal{M}$.
A poisson process $P_λ$ on $\mathbb{R}^{d}$ with causal structure inherited from the the usual Minkowski metric on $\mathbb{R}^{d}$ has a normalised discrete causal distance $D_λ(x,y)$ given by the height of the longest causal chain normalised by $λ^{1/d}c_{d}$. We prove that $P_λ$ restricted to a compact set $Q$ converges in probability in the sense of Noldus to $Q$ with the Minkowksi metric.
We prove that uniqueness for the Calderón problem on a Riemannian manifold with boundary follows from a hypothetical unique continuation property for the elliptic operator $Δ+V+(Λ^{1}_{t}-q)\otimes (Λ^{2}_{t}-q)$ defined on $\partial\mathcal{M}^{2}\times [0,1]$ where $V$ and $q$ are potentials and $Λ^{i}_{t}$ is a Dirichlet-Neumann operator at depth $t$. This is done by showing that the difference of two Dirichlet-Neumann maps is equal to the Neumann boundary values of the solution to an inhomogeneous equation for said operator, where the source term is a measure supported on the diagonal of $\partial\mathcal{M}^{2}$.
The central theme in this paper is the Hopf-Laplace equation, which represents stationary solutions with respect to the inner variation of the Dirichlet integral. Among such solutions are harmonic maps. Nevertheless, minimization of the Dirichlet energy among homeomorphisms often leads to mappings which are neither harmonic nor homeomorphisms. We prove that such mappings are harmonic outside of a singular set with small image. On the singular set they are locally Lipschitz, but not necessarily differentiable.