Boundary regularity for flows of nonparametric surfaces driven by mean curvature

2012 ◽  
Author(s):  
Nina N. Uraltseva
Keyword(s):  
Mean Curvature ◽  
Boundary Regularity

scholarly journals Boundary behaviour of solutions of the non-parametric least area problem

1982 ◽  
Vol 26 (1) ◽  
pp. 17-27 ◽  
Author(s):  
Leon Simon
Keyword(s):  
Mean Curvature ◽  
Boundary Data ◽  
Boundary Regularity ◽  
Regularity Of Solutions ◽  
Sufficient Condition ◽  
Area Problem ◽  
The Mean ◽  
The Given ◽  
Least Area ◽  
Non Parametric

Previous work concerning boundary regularity of solutions of the non-parametric least area problem leaves open the question of regularity of solutions at points where the mean curvature of the boundary of the domain vanishes. We here prove that the solutions may be discontinuous at such points, even when the given boundary data is smooth. We also give a sufficient condition which will ensure continuity at such points.

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.

scholarly journals An Analog of the Neumann Problem for the 1-Laplace Equation in the Metric Setting: Existence, Boundary Regularity, and Stability

2018 ◽  
Vol 6 (1) ◽  
pp. 1-31 ◽  
Author(s):  
Panu Lahti ◽  
Lukáš Malý ◽  
Nageswari Shanmugalingam
Keyword(s):  
Mean Curvature ◽  
Boundary Data ◽  
Metric Spaces ◽  
Neumann Boundary ◽  
Boundary Regularity ◽  
Characteristic Functions ◽  
Bounded Domains ◽  
Neumann Boundary Value Problem ◽  
The Neumann Problem ◽  
The Stability

Abstract We study an inhomogeneous Neumann boundary value problem for functions of least gradient on bounded domains in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show that solutions exist under certain regularity assumptions on the domain, but are generally nonunique. We also show that solutions can be taken to be differences of two characteristic functions, and that they are regular up to the boundary when the boundary is of positive mean curvature. By regular up to the boundary we mean that if the boundary data is 1 in a neighborhood of a point on the boundary of the domain, then the solution is −1 in the intersection of the domain with a possibly smaller neighborhood of that point. Finally, we consider the stability of solutions with respect to boundary data.

scholarly journals On obstacle problem for mean curvature flow with driving force

2019 ◽  
Vol 4 (1) ◽  
pp. 9-29
Author(s):  
Yoshikazu Giga ◽  
Hung V. Tran ◽  
Longjie Zhang
Keyword(s):  
Mean Curvature ◽  
Driving Force ◽  
Obstacle Problem ◽  
Mean Curvature Flow ◽  
Boundary Regularity ◽  
Curvature Flow ◽  
Convergence Result ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Full Characterization

Abstract In this paper, we study an obstacle problem associated with the mean curvature flow with constant driving force. Our first main result concerns interior and boundary regularity of the solution. We then study in details the large time behavior of the solution and obtain the convergence result. In particular, we give full characterization of the limiting profiles in the radially symmetric setting.

Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6449-6459 ◽  
Author(s):  
Akram Ali ◽  
Siraj Uddin ◽  
Wan Othman ◽  
Cenap Ozel
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Warped Products ◽  
Equality Case ◽  
Cosymplectic Manifolds ◽  
Almost Cosymplectic Manifold ◽  
Locally Conformal ◽  
Totally Real ◽  
Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

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