scholarly journals Holomorphic triples and the prescribed curvature problem on $S^2$

2016 ◽  
Vol 24 (3) ◽  
pp. 559-591
Author(s):  
Alexandre C. Gonçalves
Keyword(s):  
Prescribed Curvature ◽  
Curvature Problem

OPTIMAL INTERFACE ERROR ESTIMATES FOR A DISCRETE DOUBLE OBSTACLE APPROXIMATION TO THE PRESCRIBED CURVATURE PROBLEM

1999 ◽  
Vol 09 (06) ◽  
pp. 799-823 ◽  
Author(s):  
BARBARA CECON ◽  
MAURIZIO PAOLINI ◽  
MARIANGELA ROMEO
Keyword(s):  
Error Estimate ◽  
Piecewise Linear ◽  
Mesh Size ◽  
Perturbation Parameter ◽  
Element Discretization ◽  
The Maximum Principle ◽  
Formal Asymptotics ◽  
Prescribed Curvature ◽  
The Second Variation ◽  
Curvature Problem

In this paper we consider the so-called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend Ref. 10 with the introduction of the same nonregular potential used for the evolution problem in Ref. 9 and prove an optimal [Formula: see text] error estimate for nondegenerate minimizers (where ε represents the perturbation parameter). Following Ref. 10 the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error estimate for discrete minima using the Rannacher–Scott L∞-estimates and under appropriate restrictions on the mesh size ([Formula: see text] with σ>5/2).

scholarly journals Nonlinear Partial Differential Equations Arising from Prescribed Curvature Problems

2020 ◽  
pp. 84-101
Author(s):  
Man Chun Leung
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Scalar Curvature ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Pde ◽  
Minkowski Problem ◽  
Partial Differential ◽  
Prescribed Curvature ◽  
Nirenberg Problem ◽  
Curvature Problem

We consider two prominent nonlinear partial differential equations (nonlinear PDE) linked to the prescribed curvature problems, namely, the Minkowski problem and the KazdanWarner/Nirenberg problem (prescribed scalar curvature problem). This article addresses some of the modern techniques in analysis used to draw out a number of the profound features in these equations.

Vortices and the prescribed curvature problem

1985 ◽  
Vol 18 (11) ◽  
pp. L637-L641 ◽  
Author(s):  
A Comtet ◽  
P J Houston
Keyword(s):  
Prescribed Curvature ◽  
Curvature Problem

scholarly journals Yamabe type equations with a sign-changing nonlinearity, and the prescribed curvature problem

2016 ◽  
Vol 260 (10) ◽  
pp. 7416-7497 ◽  
Author(s):  
Bruno Bianchini ◽  
Luciano Mari ◽  
Marco Rigoli
Keyword(s):  
Prescribed Curvature ◽  
Curvature Problem

Classical and Non-Classical Sign-Changing Solutions of a One-Dimensional Autonomous Prescribed Curvature Equation

2007 ◽  
Vol 7 (4) ◽  
Author(s):  
Franco Obersnel
Keyword(s):  
One Dimensional ◽  
Multiplicity Of Solutions ◽  
Curvature Equation ◽  
Classical Sign ◽  
Existence And Multiplicity ◽  
Prescribed Curvature ◽  
The One ◽  
Sign Changing Solutions ◽  
Curvature Problem

AbstractWe discuss existence and multiplicity of solutions of the one-dimensional autonomous prescribed curvature problemdepending on the behaviour at the origin and at infinity of the function f. We consider solutions that are possibly discontinuous at the points where they attain the value zero.

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