scholarly journals Nonlinear partial differential equations and applications: Some nonlinear elliptic equations from geometry

2002 ◽  
Vol 99 (24) ◽  
pp. 15287-15290 ◽  
Author(s):  
Y. Li
Author(s):  
Lawrence C. Evans

SynopsisWe demonstrate how a fairly simple “perturbed test function” method establishes periodic homogenisation for certain Hamilton-Jacobi and fully nonlinear elliptic partial differential equations. The idea, following Lions, Papanicolaou and Varadhan, is to introduce (possibly nonsmooth) correctors, and to modify appropriately the theory of viscosity solutions, so as to eliminate then the effects of high-frequency oscillations in the coefficients.


2004 ◽  
Vol 4 (3) ◽  
Author(s):  
David Hartenstine ◽  
Klaus Schmitt

AbstractThere are many notions of solutions of nonlinear elliptic partial differential equations. This paper is concerned with solutions which are obtained as suprema (or infima) of so-called subfunctions (superfunctions) or viscosity subsolutions (viscosity supersolutions). The paper also explores the relationship of these (generalized) solutions of differential inequalities and provides a relevant example for which existence questions have been studied using these concepts.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.


Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.


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