Some maximum principles for nonlinear elliptic equations in divergence form with applications to capillary surfaces and to surfaces of constant mean curvature

1979 ◽  
Vol 3 (2) ◽  
pp. 193-211 ◽  
Author(s):  
L.E. Payne ◽  
G.A. Philippin
Keyword(s):  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Maximum Principles ◽  
Capillary Surfaces ◽  
Nonlinear Elliptic

Elliptic equations on manifolds and isoperimetric inequalities

1990 ◽  
Vol 114 (3-4) ◽  
pp. 213-227 ◽  
Author(s):  
Andrea Cianchi
Keyword(s):  
Riemannian Manifolds ◽  
Elliptic Equations ◽  
A Priori ◽  
Neumann Boundary ◽  
Nonlinear Elliptic Equations ◽  
Isoperimetric Inequalities ◽  
Divergence Form ◽  
Sharp Estimates ◽  
Nonlinear Elliptic ◽  
Homogeneous Neumann Boundary Condition

SynopsisWe consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.

scholarly journals Uniqueness of weak solution for nonlinear elliptic equations in divergence form

2000 ◽  
Vol 23 (5) ◽  
pp. 313-318 ◽  
Author(s):  
Xu Zhang
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Uniqueness Result ◽  
Quasilinear Elliptic Equations ◽  
Nonlinear Elliptic ◽  
Quasilinear Elliptic

We study the uniqueness of weak solutions for quasilinear elliptic equations in divergence form. Some counterexamples are given to show that our uniqueness result cannot be improved in the general case.

scholarly journals Besov regularity for solutions of p-harmonic equations

2017 ◽  
Vol 8 (1) ◽  
pp. 762-778 ◽  
Author(s):  
Albert Clop ◽  
Raffaella Giova ◽  
Antonia Passarelli di Napoli
Keyword(s):  
Elliptic Equations ◽  
Lipschitz Space ◽  
Nonlinear Elliptic Equations ◽  
Divergence Form ◽  
Type Operator ◽  
Differentiability Property ◽  
Nonlinear Elliptic ◽  
Besov Regularity ◽  
Fractional Differentiability

Abstract We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., {\operatorname{div}\mathcal{A}(x,Du)=\operatorname{div}F,} when {\mathcal{A}} is a p-harmonic type operator, and under the assumption that {x\mapsto\mathcal{A}(x,\xi\/)} belongs to the critical Besov–Lipschitz space {B^{\alpha}_{{n/\alpha},q}} . We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When {\operatorname{div}F=0} , we show that an analogous extra differentiability property for Du holds true under a Triebel–Lizorkin assumption on the partial map {x\mapsto\mathcal{A}(x,\xi\/)} .

Sign in / Sign up

Export Citation Format

Share Document