Search results for "PDE"
showing 10 items of 558 documents
Numerical Study of breakup in generalized Korteweg-de Vries and Kawahara equations
2010
This article is concerned with a conjecture by one of the authors on the formation of dispersive shocks in a class of Hamiltonian dispersive regularizations of the quasilinear transport equation. The regularizations are characterized by two arbitrary functions of one variable, where the condition of integrability implies that one of these functions must not vanish. It is shown numerically for a large class of equations that the local behaviour of their solution near the point of gradient catastrophe for the transport equation is described locally by a special solution of a Painlev\'e-type equation. This local description holds also for solutions to equations where blow up can occur in finit…
A remark on the radial minimizer of the Ginzburg-Landau functional
2014
Let Omega subset of R-2 be a bounded domain with the same area as the unit disk B-1 and letE-epsilon(u, Omega) = 1/2 integral(Omega) vertical bar del u vertical bar(2) dx + 1/4 epsilon(2) integral(Omega) (vertical bar u vertical bar(2) - 1)(2) dxbe the Ginzburg-Landau functional. Denote by (u) over tilde (epsilon) the radial solution to the Euler equation associated to the problem min {E-epsilon (u, B-1) : u vertical bar(partial derivative B1) = x} and byK = {v = (v(1), v(2)) is an element of H-1 (Omega; R-2) : integral(Omega) v(1) dx = integral(Omega) v(2) dx = 0,integral(Omega) vertical bar v vertical bar(2) dx >= integral(B1) vertical bar(u) over tilde vertical bar(2) dx}.In this note…
Boundary regularity for degenerate and singular parabolic equations
2013
We characterise regular boundary points of the parabolic $p$-Laplacian in terms of a family of barriers, both when $p>2$ and $1<p<2$. Due to the fact that $p\not=2$, it turns out that one can multiply the $p$-Laplace operator by a positive constant, without affecting the regularity of a boundary point. By constructing suitable families of barriers, we give some simple geometric conditions that ensure the regularity of boundary points.
A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group
2017
We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group $\mathbb{H}_1$. Several auxiliary properties of quasiconformal mappings between subdomains of $\mathbb{H}_1$ are proven, including distortion of balls estimates and local BMO-estimates for the logarithm of the Jacobian of a quasiconformal mapping. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in $\mathbb{H}_1$. The theorems are discussed for…
Short time existence of the classical solution to the fractional mean curvature flow
2019
Abstract We establish short-time existence of the smooth solution to the fractional mean curvature flow when the initial set is bounded and C 1 , 1 -regular. We provide the same result also for the volume preserving fractional mean curvature flow.
Local Gauge Conditions for Ellipticity in Conformal Geometry
2013
In this article we introduce local gauge conditions under which many curvature tensors appearing in conformal geometry, such as the Weyl, Cotton, Bach, and Fefferman-Graham obstruction tensors, become elliptic operators. The gauge conditions amount to fixing an $n$-harmonic coordinate system and normalizing the determinant of the metric. We also give corresponding elliptic regularity results and characterizations of local conformal flatness in low regularity settings.
Inverse problems for elliptic equations with fractional power type nonlinearities
2020
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
On Radon Transforms on Tori
2014
We show injectivity of the X-ray transform and the $d$-plane Radon transform for distributions on the $n$-torus, lowering the regularity assumption in the recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of the X-ray transform on the $n$-torus for tensor fields of any order, allowing the tensors to have distribution valued coefficients. These imply new injectivity results for the periodic broken ray transform on cubes of any dimension.
Invariant distributions, Beurling transforms and tensor tomography in higher dimensions
2014
In the recent articles \cite{PSU1,PSU3}, a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natur…
Optimal transport maps on Alexandrov spaces revisited
2018
We give an alternative proof for the fact that in $n$-dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely $(n-1)$-unrectifiable starting measure, and that this plan is induced by an optimal map.