scholarly journals Borderline gradient estimates at the boundary in Carnot groups

2020 ◽  
pp. 1-34
Author(s):  
Ramesh Manna ◽  
Ram Baran Verma
Keyword(s):  
Lorentz Space ◽  
Linear Equations ◽  
Regularity Result ◽  
Boundary Regularity ◽  
Gradient Estimates ◽  
Carnot Groups ◽  
Horizontal Gradient ◽  
Homogeneous Dimension

In this article, we prove the continuity of the horizontal gradient near a C1,Dini non-characteristic portion of the boundary for solutions to $\Gamma ^{0,{\rm Dini}}$ perturbations of horizontal Laplaceans as in (1.1) below, where the scalar term is in scaling critical Lorentz space L(Q, 1) with Q being the homogeneous dimension of the group. This result can be thought of both as a sharpening of the $\Gamma ^{1,\alpha }$ boundary regularity result in [4] as well as a subelliptic analogue of the main result in [1] restricted to linear equations.

scholarly journals A two-weight Sobolev inequality for Carnot-Carathéodory spaces

2020 ◽  
Author(s):  
Angela Alberico ◽  
Patrizia Di Gironimo
Keyword(s):  
Finite Rank ◽  
Sobolev Inequality ◽  
Vector Fields ◽  
Horizontal Gradient ◽  
Smooth Vector ◽  
Rank Condition ◽  
Homogeneous Dimension

Abstract Let $$X = \{X_1,X_2, \ldots ,X_m\}$$ X = { X 1 , X 2 , … , X m } be a system of smooth vector fields in $${{\mathbb R}^n}$$ R n satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space $$\mathbb G$$ G associated to system X$$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R}\frac{1}{K(x)} \; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2}, \end{aligned}$$ 1 ∫ B R K ( x ) d x ∫ B R | u | t K ( x ) d x 1 / t ≤ C R 1 ∫ B R 1 K ( x ) d x ∫ B R | X u | 2 K ( x ) d x 1 / 2 , where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt’s class $$A_2$$ A 2 and Gehring’s class $$G_{\tau }$$ G τ , where $$\tau $$ τ is a suitable exponent related to the homogeneous dimension.

scholarly journals A boundary regularity result for minimizers of variational integrals with nonstandard growth

2018 ◽  
Vol 177 ◽  
pp. 153-168
Author(s):  
Miroslav Bulíček ◽  
Erika Maringová ◽  
Bianca Stroffolini ◽  
Anna Verde
Keyword(s):  
Regularity Result ◽  
Boundary Regularity ◽  
Nonstandard Growth ◽  
Variational Integrals

scholarly journals Hausdorff measure of the singular set of quasiregular maps on Carnot groups

2003 ◽  
Vol 2003 (35) ◽  
pp. 2203-2220 ◽  
Author(s):  
Irina Markina
Keyword(s):  
Vector Space ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Quasiregular Mapping ◽  
Carnot Groups ◽  
Singular Set ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Dimensional Hausdorff Measure ◽  
Homogeneous Dimension

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Letνstand for the homogeneous dimension of a Carnot group and letmbe the index of the last vector space of the corresponding Lie algebra. We prove that the(ν−m−1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimates of the local index of quasiregular mappings are also obtained.

scholarly journals Starshapedeness for fully non‐linear equations in Carnot groups

2018 ◽  
Vol 99 (3) ◽  
pp. 901-918 ◽  
Author(s):  
Federica Dragoni ◽  
Nicola Garofalo ◽  
Paolo Salani
Keyword(s):  
Linear Equations ◽  
Carnot Groups ◽  
Non Linear ◽  
Non Linear Equations

scholarly journals Subdivisions of Ring Dupin Cyclides Using Bézier Curves with Mass Points

2021 ◽  
Vol 20 ◽  
pp. 581-596
Author(s):  
Lionel Garnier ◽  
Lucie Druoton ◽  
Jean-Paul Bécar ◽  
Laurent Fuchs ◽  
Géraldine Morin
Keyword(s):  
Lorentz Space ◽  
Geometric Modeling ◽  
Linear Equations ◽  
Algebraic Surfaces ◽  
Bezier Curves ◽  
Bézier Curves ◽  
French Mathematician ◽  
Cad Cam ◽  
Dupin Cyclide ◽  
Dupin Cyclides

Dupin cyclides are algebraic surfaces introduced for the first time in 1822 by the French mathematician Pierre-Charles Dupin. A Dupin cyclide can be defined as the envelope of a one-parameter family of oriented spheres, in two different ways. R. Martin is the first author who thought to use these surfaces in CAD/CAM and geometric modeling. The Minkowski-Lorentz space is a generalization of the space-time used in Einstein’s theory, equipped of the non-degenerate indefinite quadratic form QM(u) = x^2 + y^2 + z^2 - c^2 t^2where (x, y, z) are the spacial components of the vector u and t is the time component of u and c is the constant of the speed of light. In this Minkowski-Lorentz space, a Dupin cyclide is the union of two conics on the unit pseudo-hypersphere, called the space of spheres, and a singular point of a Dupin cyclide is represented by an isotropic vector. Then, we model Dupin cyclides using rational quadratic Bézier curves with mass points. The subdivisions of a surface i.e. a Dupin cyclide, is equivalent to subdivide two curves of degree 2, independently, whereas in the 3D Euclidean space ε3, the same work implies the subdivision of a rational quadratic Bézier surface and resolutions of systems of three linear equations. The first part of this work is to consider ring Dupin cyclides because the conics are circles which look like ellipses.

scholarly journals Global boundedness, interior gradient estimates, and boundary regularity for the mean curvature equation with boundary conditions

2004 ◽  
Vol 2004 (18) ◽  
pp. 913-948
Author(s):  
Fei-Tsen Liang
Keyword(s):  
Dirichlet Problem ◽  
Mean Curvature ◽  
Boundary Regularity ◽  
Gradient Estimates ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
Global Estimates ◽  
The Mean ◽  
Boundary Contact ◽  
Global Boundedness

We obtain global estimates for the modulus, interior gradient estimates, and boundary Hölder continuity estimates for solutionsuto the capillarity problem and to the Dirichlet problem for the mean curvature equation merely in terms of the mean curvature, together with the boundary contact angle in the capillarity problem and the boundary values in the Dirichlet problem.

Analysis of temperatures in sedimentary basins: the Michigan Basin

Geophysics ◽  
1985 ◽  
Vol 50 (8) ◽  
pp. 1318-1334 ◽  
Author(s):  
Marvin A. Speece ◽  
Timothy D. Bowen ◽  
James L. Folcik ◽  
Henry N. Pollack
Keyword(s):  
Linear Equations ◽  
Synthetic Data ◽  
Sedimentary Basins ◽  
Negative Temperature ◽  
Temperature Gradients ◽  
Gradient Estimates ◽  
Data Sets ◽  
Michigan Basin ◽  
Crust And Upper Mantle ◽  
The Difference

We develop an analytical and numerical methodology for the analysis of large bottom‐hole temperature (BHT) data sets from sedimentary basins, and test the methodology using temperature, stratigraphic, and lithologic data from 411 boreholes in the Michigan Basin. Least‐squares estimates of temperature gradients in the formations and lithologies present are calculated as solutions to a large system of linear equations. At each borehole the temperature difference between the bottom and top of the hole is represented as a sum of temperature increments through the various formations or lithologies penetrated by the borehole. Quadratic programming techniques enable bounds to be placed on the gradient solutions in order to suppress or exclude improbable gradient estimates. Numerical experiments with synthetic data reveal that the estimates of temperature gradients for a given formation or lithology are sensitive to the degree of representation of that unit; well represented units have more stable gradient estimates in the presence of noise than do poorly represented units. The estimates of temperature gradients obtained for lithologies are more stable than those for formations and are believed to be good estimates of actual lithologic temperature gradients in the Michigan Basin. Formation temperature gradients obtained as a weighted sum of the estimates of the component lithologic temperature gradients appear to be good estimates of the average temperature gradients for the formations of the basin. At each borehole a temperature residual exists corresponding to the difference between the observed BHT and the BHT predicted by the estimated interval temperature gradients. Residuals are far more stable than estimated temperature gradients. The values of residuals change little regardless of whether lithology, formation, bounded, or unbounded gradient estimates are used to calculate them. Maps of residuals indicate well‐defined and spatially coherent patterns of positive and negative temperature residuals. Filtered subsets of large‐magnitude residuals alone show a pattern of negative residuals coinciding with the mid‐Michigan gravity high, a geophysical feature thought to delineate a Precambrian (Keweenawan) rift zone in the crust beneath the basin. Thermal models of the Michigan Basin and the crust and upper mantle beneath the basin indicate that the suspected rift beneath the basin can cause a variation in basement heat flow sufficient to produce temperature residuals of the magnitude observed in the sediments, with negative temperature residuals over the area of the rift.

scholarly journals Regularity Result for Quasilinear Elliptic Systems with Super Quadratic Natural Growth Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Shuhong Chen ◽  
Zhong Tan
Keyword(s):  
Growth Condition ◽  
Elliptic Systems ◽  
Regularity Result ◽  
Boundary Regularity ◽  
Quadratic Growth ◽  
Natural Growth ◽  
General Criterion ◽  
Natural Growth Condition ◽  
Quasilinear Elliptic Systems ◽  
Quasilinear Elliptic

We consider boundary regularity for weak solutions of second-order quasilinear elliptic systems under natural growth condition with super quadratic growth and obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. Combined with existing results on interior partial regularity, this result yields an upper bound on the Hausdorff dimension of the singular set at the boundary.

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