Search results for "PDE"
showing 10 items of 558 documents
Stationary sets and asymptotic behavior of the mean curvature flow with forcing in the plane
2020
We consider the flat flow solutions of the mean curvature equation with a forcing term in the plane. We prove that for every constant forcing term the stationary sets are given by a finite union of disks with equal radii and disjoint closures. On the other hand for every bounded forcing term tangent disks are never stationary. Finally in the case of an asymptotically constant forcing term we show that the only possible long time limit sets are given by disjoint unions of disks with equal radii and possibly tangent. peerReviewed
An inverse problem for the minimal surface equation
2022
We use the method of higher order linearization to study an inverse boundary value problem for the minimal surface equation on a Riemannian manifold $(\mathbb{R}^n,g)$, where the metric $g$ is conformally Euclidean. In particular we show that with the knowledge of Dirichlet-to-Neumann map associated to the minimal surface equation, one can determine the Taylor series of the conformal factor $c(x)$ at $x_n=0$ up to a multiplicative constant. We show this both in the full data case and in some partial data cases.
A sharp stability estimate for tensor tomography in non-positive curvature
2021
Funder: University of Cambridge
Stationary sets of the mean curvature flow with a forcing term
2020
We consider the flat flow approach for the mean curvature equation with forcing in an Euclidean space $\mathbb R^n$ of dimension at least 2. Our main results states that tangential balls in $\mathbb R^n$ under any flat flow with a bounded forcing term will experience fattening, which generalizes the result by Fusco, Julin and Morini from the planar case to higher dimensions. Then, as in the planar case, we are able to characterize stationary sets in $\mathbb R^n$ for a constant forcing term as finite unions of equisized balls with mutually positive distance.
Volume preserving mean curvature flows near strictly stable sets in flat torus
2021
In this paper we establish a new stability result for the smooth volume preserving mean curvature flow in flat torus $\mathbb T^n$ in low dimensions $n=3,4$. The result says roughly that if the initial set is near to a strictly stable set in $\mathbb T^n$ in $H^3$-sense, then the corresponding flow has infinite lifetime and converges exponentially fast to a translate of the strictly stable (critical) set in $W^{2,5}$-sense.
C1,α-regularity for variational problems in the Heisenberg group
2017
We study the regularity of minima of scalar variational integrals of $p$-growth, $1<p<\infty$, in the Heisenberg group and prove the H\"older continuity of horizontal gradient of minima.
Nonlinear Liouville Problems in a Quarter Plane
2016
We answer affirmatively the open problem proposed by Cabr\'e and Tan in their paper "Positive solutions of nonlinear problems involving the square root of the Laplacian" (see Adv. Math. {\bf 224} (2010), no. 5, 2052-2093).
Asymptotic $C^{1,γ}$-regularity for value functions to uniformly elliptic dynamic programming principles
2022
In this paper we prove an asymptotic C1,γ-estimate for value functions of stochastic processes related to uniformly elliptic dynamic programming principles. As an application, this allows us to pass to the limit with a discrete gradient and then to obtain a C1,γ-result for the corresponding limit PDE. peerReviewed
Multi-marginal entropy-transport with repulsive cost
2020
In this paper we study theoretical properties of the entropy-transport functional with repulsive cost functions. We provide sufficient conditions for the existence of a minimizer in a class of metric spaces and prove the $\Gamma$-convergence of the entropy-transport functional to a multi-marginal optimal transport problem with a repulsive cost. We also prove the entropy-regularized version of the Kantorovich duality.
Asymptotic Lipschitz regularity for tug-of-war games with varying probabilities
2018
We prove an asymptotic Lipschitz estimate for value functions of tug-of-war games with varying probabilities defined in $\Omega\subset \mathbb R^n$. The method of the proof is based on a game-theoretic idea to estimate the value of a related game defined in $\Omega\times \Omega$ via couplings.