Search results for "Mathematical analysis"
showing 10 items of 2409 documents
Historischer Überblick zur mathematischen Theorie von Unstetigkeitswellen seit Riemann und Christoffel
1981
We give a brief historical account of the development of the mathematical theory of propagation of discontinuities in gases, fluids or elastic materials. The theory was initiated by Riemann who investigated the propagation of shocks in one-dimensional isentropic gas flow. Riemann’s results were used by Christoffel to treat, more generally, the propagation of (first order) discontinuity surfaces in three-dimensional flows of perfect fluids. Subsequently Christoffel applied his general theory to first order waves in certain elastic materials. Independently of Riemann and Christoffel significant contributions were made by Hugoniot. The theory was completed in Hadamard’s celebrated monograph [3…
Corrigendum to ``A smooth foliation of the 5-sphere by complex surfaces"
2011
International audience
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.
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.
Semianalyticity of isoperimetric profiles
2009
It is shown that, in dimensions $<8$, isoperimetric profiles of compact real analytic Riemannian manifolds are semi-analytic.
Geodesic ray transform with matrix weights for piecewise constant functions
2019
We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three and higher we assume a foliation condition. We make no assumption regarding conjugate points or differentiability of the weight. This extends recent results for unweighted transforms.
X-ray Tomography of One-forms with Partial Data
2021
If the integrals of a one-form over all lines meeting a small open set vanish and the form is closed in this set, then the one-form is exact in the whole Euclidean space. We obtain a unique continuation result for the normal operator of the X-ray transform of one-forms, and this leads to one of our two proofs of the partial data result. Our proofs apply to compactly supported covector-valued distributions.
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.
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.
The Riemannian manifold of all Riemannian metrics
1991
In this paper we study the geometry of (M, G) by using the ideas developed in [Michor, 1980]. With that differentiable structure on M it is possible to use variational principles and so we start in section 2 by computing geodesics as the curves in M minimizing the energy functional. From the geodesic equation, the covariant derivative of the Levi-Civita connection can be obtained, and that provides a direct method for computing the curvature of the manifold. Christoffel symbol and curvature turn out to be pointwise in M and so, although the mappings involved in the definition of the Ricci tensor and the scalar curvature have no trace, in our case we can define the concepts of ”Ricci like cu…