Regular Hypersurface Research Articles

Schauder estimates for parabolic equations with degenerate or singular weights

We establish some C0,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{0,\\alpha }$$\\end{document} and C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} as a power a>-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a > -1$$\\end{document} of the distance to Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document}. The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar C1,α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\alpha }$$\\end{document} estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

On the Compact Integral Hypersurfaces of the Multidimensional Homogeneous Darboux System

The ordinary at $m=1$ and the completely solvable at $m>1$ the real homogeneous Darboux systems $$dx_i=\sum\limits_{j=1}^{m}\Bigl(L_{ij}(x)+x_iP_j(x) \Bigr)dt_j, i=\overline{1,n},$$ where $n>1,\ m<n,\ L_{ij}(x)=\sum\limits_{k=1}^{n}a_{ijk}x_k,\ a_{ijk}\in\mathbb{R},\ k=\overline{1,n},\ j=\overline{1,m},\ i=\overline{1,n},\ P_j(x),\ j=\overline{1,m}$, are twice smooth homogeneous functions of order $\rho\geqslant 1$ and the rank of the matrix corresponding to the right-hand sides of the Darboux systems is equal to m almost everywhere on $\mathbb{R}^n$, are considered. The concept of a regular integral hypersurface of these systems is introduced. It is proved that the systems under consideration cannot have more than one isolated compact regular integral hypersurface. An example of a system with an isolated compact regular integral hypersurface is given.

On the development of shocks in fluids

In my 2007 work, I studied the formation of shocks in the context of Eulerian equations of the mechanics of compressible fluids. That work studied the maximal classical development of smooth initial data. The present review article is a presentation of my 2019 work, which addresses the problem of the physical continuation of the solution past the point of shock formation. The problem requires the construction of a hypersurface, the shock hypersurface, originating from a given singular spacelike surface, which is acoustically spacelike as viewed from its past, and the construction of a new solution in the future of the union of a given regular null hypersurface with the shock hypersurface, a solution which extends continuously the prior maximal classical solution across the given regular null hypersurface, but which displays discontinuities in physical variables across the shock hypersurface in accordance with the mass, momentum, and energy conservation laws. Moreover, the shock hypersurface is to be acoustically timelike as viewed from its future. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a first order quasilinear hyperbolic system of PDE, with characteristic initial data, which are singular at the past boundary of the initial null hypersurface, that boundary being the singular surface of origin. My 2019 work solved a restricted form of the problem through the introduction of new geometric and analytic methods. This Review is focused on the presentation of these methods.

The Dirichlet Problem for a Class of Prescribed Curvature Equations

In this paper, we consider the Dirichlet problem for a class of prescribed curvature equations. Both degenerate and non-degenerate cases are considered. The existence of the \(C^{1,1}\) regular graphic hypersurfaces with prescribing a class of curvatures and constant boundary is proved for the degenerate case.

Global Regular Null Hypersurfaces in a Perturbed Schwarzschild Black Hole Exterior

The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole. A significant point of the result is that the ambient spacetime metric is not required to be differentiable in all directions.

On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

Doubling coverings via resolution of singularities and preparation

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

The surface diffusion and the Willmore flow for uniformly regular hypersurfaces

We consider the surface diffusion and Willmore flows acting on a general class of (possibly non–compact) hypersurfaces parameterized over a uniformly regular reference manifold possessing a tubular neighborhood with uniform radius. The surface diffusion and Willmore flows each give rise to a fourth–order quasilinear parabolic equation with nonlinear terms satisfying a specific singular structure. We establish well–posedness of both flows for initial surfaces that are $ C^{1+\alpha} $–regular and parameterized over a uniformly regular hypersurface. For the Willmore flow, we also show long–term existence for initial surfaces which are $ C^{1+\alpha} $–close to a sphere, and we prove that these solutions become spherical as time goes to infinity.

Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

Abstract Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$ such that each manifold $Z_d$ is diffeomorphic to a component of the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\Gamma$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\Gamma$ is bounded by a polynomial in $\deg (p)$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$ containing a subset $D$ homeomorphic to a disk, and a family of polynomials $\{p_m\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$ i.e. the zero set of $p_m$ in $D$ is isotopic to $Z_{d_m}$ in ${\mathbb{R}}^{n-1}$. This says that, up to extracting subsequences, the intersection of $\Gamma$ with a hypersurface of degree $d$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $0 \leq k \leq n-2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact semianalytic hypersurface $\Gamma \subset {\mathbb{R}}\textrm{P}^n$, a subsequence $\{a_{d_m}\}_{m\in \mathbb{N}}$ and homogeneous polynomials $\{p_{m}\}_{m\in \mathbb{N}}$ of degree $\deg (p_m)=d_m$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $b_k$ denotes the $k$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $\Gamma$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $d$, of the set $\Sigma _{d_m,a, \Gamma }$ of polynomials verifying (0.1) is positive, but there exists a constant $c_\Gamma$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $a$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $\Gamma$, for most polynomials a Bézout-type bound holds for the intersection $\Gamma \cap Z(p)$: for every $0\leq k\leq n-2$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$

Birational geometry of singular Fano hypersurfaces of index two

For a Zariski general (regular) hypersurface V of degree M in the (M+1)-dimensional projective space, where Mgeqslant 16, with at most quadratic singularities of rank geqslant 13, we give a complete description of the structures of rationally connected (or Fano-Mori) fibre space: every such structure over a positive-dimensional base is a pencil of hyperplane sections. This implies, in particular, that V is non-rational and its groups of birational and biregular automorphisms coincide: mathrm{Bir} V = mathrm{Aut} V. The set of non-regular hypersurfaces has codimension at least frac{1}{2}(M-11)(M-10)-10 in the natural parameter space.

Affine focal sets of codimension-2 submanifolds contained in hypersurfaces

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

Complete conformal classification of the Friedmann–Lemaître–Robertson–Walker solutions with a linear equation of state

We completely classify Friedmann–Lemaître–Robertson–Walker solutions with spatial curvature and equation of state , according to their conformal structure, singularities and trapping horizons. We do not assume any energy conditions and allow , thereby going beyond the usual well-known solutions. For each spatial curvature, there is an initial spacelike big-bang singularity for w > −1/3 and , while there is no big-bang singularity for w < −1 and . For K = 0 or −1, −1 < w < −1/3 and , there is an initial null big-bang singularity. For each spatial curvature, there is a final spacelike future big-rip singularity for w < −1 and , with null geodesics being future complete for but incomplete for w < −5/3. For w = −1/3, the expansion speed is constant. For −1 < w < −1/3 and K = 1, the universe contracts from infinity, then bounces and expands back to infinity. For K = 0, the past boundary consists of timelike infinity and a regular null hypersurface for −5/3 < w < −1, while it consists of past timelike and past null infinities for . For w < −1 and K = 1, the spacetime contracts from an initial spacelike past big-rip singularity, then bounces and blows up at a final spacelike future big-rip singularity. For w < −1 and K = −1, the past boundary consists of a regular null hypersurface. The trapping horizons are timelike, null and spacelike for , and , respectively. A negative energy density () is possible only for K = −1. In this case, for w > −1/3, the universe contracts from infinity, then bounces and expands to infinity; for −1 < w < −1/3, it starts from a big-bang singularity and contracts to a big-crunch singularity; for w < −1, it expands from a regular null hypersurface and contracts to another regular null hypersurface.

Nonlinear estimates for hypersurfaces in terms of their fundamental forms

A sufficiently regular hypersurface immersed in the (n+1)-dimensional Euclidean space is determined up to a proper isometry of Rn+1 by its first and second fundamental forms. As a consequence, a sufficiently regular hypersurface with boundary, whose position and positively-oriented unit normal vectors are given on a non-empty portion of its boundary, is uniquely determined by its first and second fundamental forms. We establish here stronger versions of these uniqueness results by means of inequalities showing that an appropriate distance between two immersions from a domain ω of Rn into Rn+1 is bounded by the Lp-norm of the difference between matrix fields defined in terms of the first and second fundamental forms associated with each immersion.

Regular space-like hypersurfaces in S 1 m+1 with parallel para-Blaschke tensors

In this paper, we give a complete conformal classification of the regular space-like hypersurfaces in the de Sitter Space S m+1 1 with parallel para-Blaschke tensors.

Conformal regular spacelike hypersurfaces in a conformal space $$\mathbb {Q}^{n+1}_1$$ Q 1 n + 1

Let \(x: M \rightarrow \mathbb {Q}^{n+1}_1\) be an \(n (n\ge 4)\)-dimensional conformal regular spacelike hypersurface in the conformal space \(\mathbb {Q}^{n+1}_1\) with vanishing conformal form. If x is a conformal Blaschke isoparametric spacelike hypersurface in \(\mathbb {Q}^{n+1}_1\) with three distinct conformal principal curvatures, one of which is simple or x is of parallel conformal Blaschke tensor, we obtain some classification of such conformal regular spacelike hypersurface x.

Blaschke isoparametric hypersurfaces in the conformal space ℚ 1 n+1 , I

Let x: M → Q1n+1 be a regular hypersurface in the conformal space Q1n+1. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.

Entropy conditions for scalar conservation laws with discontinuous flux revisited

We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen–Risebro–Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate.The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the “crossing condition” of Karlsen, Risebro, and Towers (2003) [31] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function.

Shades of hyperbolicity for Hamiltonians

We prove that a Hamiltonian system is globally hyperbolic if any of the following statements hold: H is robustly topologically stable; H is stably shadowable; H is stably expansive; and H has the stable weak specification property. Moreover, we prove that, for a C2-generic Hamiltonian H, the union of the partially hyperbolic regular energy hypersurfaces and the closed elliptic orbits, forms a dense subset of M. As a consequence, any robustly transitive regular energy hypersurface of a C2-Hamiltonian is partially hyperbolic. Finally, we prove that stable weakly-shadowable regular energy hypersurfaces are partially hyperbolic.

Parametrizations of level surfaces of real-valued mappings of Carnot groups

We study the regularity of the parametrizations of level surfaces of continuously horizontally differentiable real-valued mappings of Carnot groups. Equations that describe a regular hypersurface in terms of parametrizations are obtained.

RATIONAL FIRST INTEGRALS FOR POLYNOMIAL VECTOR FIELDS ON ALGEBRAIC HYPERSURFACES OF ℝn+1

Using sophisticated techniques of Algebraic Geometry, Jouanolou in 1979 showed that if the number of invariant algebraic hypersurfaces of a polynomial vector field in ℝn of degree m is at least [Formula: see text], then the vector field has a rational first integral. Llibre and Zhang used only Linear Algebra to provide a shorter and easier proof of the result given by Jouanolou. We use ideas of Llibre and Zhang to extend the Jouanolou result to polynomial vector fields defined on algebraic regular hypersurfaces of ℝn+1, this extended result completes the standard results of the Darboux theory of integrability for polynomial vector fields on regular algebraic hypersurfaces of ℝn+1.