scholarly journals Triple solutions for elliptic Dirichlet problems with a parameter

2021 ◽  
Vol 11 ◽  
pp. 100157
Author(s):  
Jian Liu ◽  
Wenguang Yu
Keyword(s):  
Dirichlet Problems

scholarly journals On the explicit representation of the trace space $$H^{\frac{3}{2}}$$ and of the solutions to biharmonic Dirichlet problems on Lipschitz domains via multi-parameter Steklov problems

2021 ◽  
Author(s):  
Pier Domenico Lamberti ◽  
Luigi Provenzano
Keyword(s):  
Fourier Series ◽  
Dirichlet Problem ◽  
Lipschitz Domain ◽  
Spectral Properties ◽  
Explicit Representation ◽  
Lipschitz Domains ◽  
Dirichlet Problems ◽  
Trace Space ◽  
In Series ◽  
Definition Of

AbstractWe consider the problem of describing the traces of functions in $$H^2(\Omega )$$ H 2 ( Ω ) on the boundary of a Lipschitz domain $$\Omega $$ Ω of $$\mathbb R^N$$ R N , $$N\ge 2$$ N ≥ 2 . We provide a definition of those spaces, in particular of $$H^{\frac{3}{2}}(\partial \Omega )$$ H 3 2 ( ∂ Ω ) , by means of Fourier series associated with the eigenfunctions of new multi-parameter biharmonic Steklov problems which we introduce with this specific purpose. These definitions coincide with the classical ones when the domain is smooth. Our spaces allow to represent in series the solutions to the biharmonic Dirichlet problem. Moreover, a few spectral properties of the multi-parameter biharmonic Steklov problems are considered, as well as explicit examples. Our approach is similar to that developed by G. Auchmuty for the space $$H^1(\Omega )$$ H 1 ( Ω ) , based on the classical second order Steklov problem.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

Filippov solutions of vector Dirichlet problems

2020 ◽  
Vol 70 (2) ◽  
pp. 401-416
Author(s):  
Hana Machů
Keyword(s):  
Differential Equations ◽  
The State ◽  
State Variables ◽  
Dirichlet Problems ◽  
Filippov Solutions ◽  
Natural Notion ◽  
Growth Restrictions ◽  
The One ◽  
The Right ◽  
Effective Growth

Abstract If in the right-hand sides of given differential equations occur discontinuities in the state variables, then the natural notion of a solution is the one in the sense of Filippov. In our paper, we will consider this type of solutions for vector Dirichlet problems. The obtained theorems deal with the existence and localization of Filippov solutions, under effective growth restrictions. Two illustrative examples are supplied.

Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior

2020 ◽  
Vol 20 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Classical Solutions ◽  
Natural Growth ◽  
Dirichlet Problems ◽  
Nonlinear Transformations

AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

A numerical algorithm for finding solutions of p-Laplacian Dirichlet problems

2007 ◽  
Vol 185 (1) ◽  
pp. 213-217
Author(s):  
G.A. Afrouzi ◽  
Z. Naghizadeh ◽  
S. Mahdavi
Keyword(s):  
Numerical Algorithm ◽  
Dirichlet Problems

Boundary value conditions for wave fronts in reaction-diffusion systems

1984 ◽  
Vol 99 (1-2) ◽  
pp. 127-136 ◽  
Author(s):  
J. M. Fraile ◽  
J. Sabina
Keyword(s):  
Wave Front ◽  
Reaction Diffusion ◽  
Propagation Direction ◽  
Dirichlet Problems ◽  
Reaction Diffusion Systems ◽  
New Class ◽  
Diffusion Systems ◽  
Front Solutions ◽  
Distinguished Boundary ◽  
Directional Wave

SynopsisIn this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.

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