Luca Capogna, Giovanna Citti, Maria Manfredini
We prove that Lipschitz intrinsic graphs in the Heisenberg groups $H^n$, with $n>1$, which are vanishing viscosity solutions of the minimal surface equation are smooth.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation ϕ_y+ [ϕ^{2}/2]_t=w, where w is a bounded function depending on ϕ.
In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.
We prove boundedness results for integral operators of fractional type and their higher order commutators between weighted spaces, including $L^p$-$L^q$, $L^p$-$BMO$ and $L^p$-Lipschitz estimates. The kernels of such operators satisfy certain size condition and a Lipschitz type regularity, and the symbol of the commutator belongs to a Lipschitz class. We also deal with commutators of fractional type operators with less regular kernels satisfying a Hörmander's type inequality. As far as we know, these last results are new even in the unweighted case. Moreover, we give a characterization result …
The goal of this article is to establish local Lipschitz continuity of weak solutions for a class of degenerated elliptic equations of divergence form, in the Heisenberg Group. The considered hypothesis for the growth and ellipticity condition is a natural generalization of the p-Laplace equation and more general quasilinear elliptic equations with polynomial or exponential type growth.
We prove local Lipschitz regularity for weak solutions to a class of degenerate parabolic PDEs modeled on the parabolic $p$-Laplacian $$\p_t u= \sum_{i=1}^{2n} X_i (|\nabla_0 u|^{p-2} X_i u),$$ in a cylinder $Ω\times \R^+$, where $Ω$ is domain in the Heisenberg group $\Hn$, and $2\le p \le 4$. The result continues to hold in the more general setting of contact sub-Riemannian manifolds.
Lipschitz regularity for solutions to an orthotropic $q$-Laplacian-type equation in the Heisenberg group
本文在Heisenberg群中研究了各向异性q-Laplacian型方程解的Lipschitz正则性，受X. Zhong工作启发，证明了解梯度的Hölder正则性，指出退化性导致在欧几里得空间中高维情况下正则性不明确。
Let D be a planar Lipschitz domain and consider the Beurling transform of the characteristic function of D, B(1_D). Let 1<p<\infty and 0<a<1 with ap>1. In this paper we show that if the outward unit normal N on bD, the boundary of D, belongs to the Besov space B_{p,p}^{a-1/p}(bD), then the Beurling transform of 1_D is in the Sobolev space W^{a,p}(D). This result is sharp. Further, together with recent results by Cruz, Mateu and Orobitg, this implies that the Beurling transform is bounded in W^{a,p}(D) if N belongs to B_{p,p}^{a-1/p}(bD), assuming that ap>2.
Lipschitz approximation of almost $\mathbb G$-perimeter minimizing boundaries in plentiful groups
本文证明丰富群中内蕴周长几乎最小化边界可由内蕴Lipschitz图逼近，首次将正则性理论推广至Heisenberg群之外。
We solve the Dirichlet problem $\left.u\right|_{\mathbb{B}^n}=φ,$ for hyperbolic Poisson's equation $Δ_h u=μ$ where $φ\in L_1(\partial \mathbb{B}^n)$ and $μ$ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson's equations, previously established at \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz's potential. Also, local Hölder continuity is proved for solution of certain hyperbolic Poisson's equations. We show that, if $u…
In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.
Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^{\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.
For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich type estimates in Lipschitz domains and Lipschitz estimates in $C^{1, α}$ domains. The $C^α$, $W^{1,p}$, and $L^p$ estimates in $C^1$ domains for systems with VMO coefficients are also studied. The appr…
The Bernstein problem for Sobolev intrinsic graphs in the Heisenberg group
该研究针对Heisenberg群中Sobolev内蕴图的Bernstein问题，证明了满足导数可积性条件的稳定次黎曼面积内蕴图必为内蕴平面，将现有Lipschitz类结果扩展到Sobolev情形。
On the Lq(Lp)-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains
研究有界Lipschitz域上带零Dirichlet边界的线性随机抛物方程在Krylov加权Sobolev空间中的存在唯一性、时间Hölder正则性，并通过嵌入定理得到Besov尺度下的正则性结果。
We present several results on the smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.
A semi-discrete scheme for horizontal mean curvature flow in the Heisenberg group
本文针对第一海森堡群的水平平均曲率流提出极小化运动半离散格式，证明了满足特定连续性条件的平坦流存在，适用于任意有限水平周长的有界集。
In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.
Let $L$ be a homogeneous sublaplacian on a 2-step stratified Lie group $G$ of topological dimension $d$ and homogeneous dimension $Q$. By a theorem due to Christ and to Mauceri and Meda, an operator of the form $F(L)$ is bounded on $L^p$ for $1 < p < \infty$ if $F$ satisfies a scale-invariant smoothness condition of order $s > Q/2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s > d/2$ is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by Müller and Stein and by Hebisch, and for the f…
Gradient estimates for an orthotropic nonlinear diffusion equation in the Heisenberg group
本文证明了 Heisenberg 群上抛物各向异性 $p$-Laplacian 型方程弱解在 $2\leq p\leq4$ 范围内的局部 Lipschitz 正则性。
We investigate observability and Lipschitz stability for the Heisenberg heat equation on the rectangular domain $$Ω= (-1,1)\times\mathbb{T}\times\mathbb{T}$$ taking as observation regions slices of the form $ω=(a,b) \times \mathbb{T} \times \mathbb{T}$ or tubes $ω= (a,b) \times ω_y \times \mathbb{T}$, with $-1<a<b<1$. We prove that observability fails for an arbitrary time $T>0$ but both observability and Lipschitz stability hold true after a positive minimal time, which depends on the distance between $ω$ and the boundary of $Ω$: $$T_{\min} \geqslant \frac{1}{8} \min\{(1+a)^2,(1-b)^2\}.$$ Our…
Lipschitz spaces adapted to Schrödinger operators on the Heisenberg group
本文针对海森堡群上的薛定谔算子$L$，引入两类适配的利普希茨空间，证明其在特定参数下等价并给出应用，丰富了相关算子的正则性研究。
We extend to the parabolic setting some of the ideas originated with Xiao Zhong's proof in \cite{Zhong} of the Hölder regularity of $p-$harmonic functions in the Heisenberg group $\Hn$. Given a number $p\ge 2$, in this paper we establish the $C^{\infty}$ smoothness of weak solutions of quasilinear pde's in $\Hn$ modelled on the equation $$\p_t u= \sum_{i=1}^{2n} X_i \bigg((1+|\nabla_0 u|^2)^{\frac{p-2}{2}} X_i u\bigg).$$
The purpose of this work is twofold. First we study the solutions of a Hamilton-Jacobi equation of the form $u_t(t,x)+\mathcal{H}(t,x,\nabla_H u(t,x))=0$, where $\nabla_H u$ represents the horizontal gradient of a function $u$ defined on the Heisenberg group ${I\!\!H}$. Motivated by the recent paper by Liu, Manfredi and Zhou (\cite{LiMaZh2016}), we prove a Lipschitz continuity preserving property for $u$ with respect to the Korányi homogeneous distances $d_G$ in ${I\!\!H}$. Secondly, we are keenly interested in introducing the game theory in ${I\!\!H}$, taking into account its Sub-Riemannian s…
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in $H^\infty$, if the coefficients enjoy enough smoothness in $x$. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysi…
In this paper we prove that every collection of measurable functions $f_α$, $|α|=m$ coincides a.e. with $m$th order derivatives of a function $g\in C^{m-1}$ whose derivatives of order $m-1$ may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
We prove that, if Δ_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-Hörmander multiplier on the positive half-line, with L^2-order of smoothness greater than n+1/2, then m(Δ_1) is L^p-bounded for 1<p<\infty. Our approach leads to an explicit description of the spectral decomposition of Δ_1 on the space of L^2-forms in terms of the spectral analysis of the sub-Laplacian L and the central derivative T, acting on scalar-valued functions.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。