scholarly journals A priori bounds for solutions of the Dirichlet problem for [Δ + λ2n(x)]u = f(x, λ) on an exterior domain

1977 ◽  
Vol 24 (3) ◽  
pp. 437-465 ◽  
Author(s):  
Clifford O Bloom ◽  
Nicholas D Kazarinoff
Keyword(s):  
Dirichlet Problem ◽  
Exterior Domain ◽  
A Priori ◽  
A Priori Bounds

scholarly journals A Priori Bounds in and in for Solutions of Elliptic Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Sara Monsurrò ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equations ◽  
A Priori ◽  
Unbounded Domains ◽  
Variational Problems ◽  
Discontinuous Coefficients ◽  
Elliptic Partial Differential Equations ◽  
Divergence Form ◽  
A Priori Bounds ◽  
A Priori Bound

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

scholarly journals Weighted Pluricomplex Energy II

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Slimane Benelkourchi
Keyword(s):  
Dirichlet Problem ◽  
Orlicz Space ◽  
Boundary Data ◽  
Borel Measure ◽  
A Priori Estimates ◽  
A Priori ◽  
Unique Function ◽  
A Priori Bounds ◽  
Priori Estimates ◽  
Monge Ampere

We continue our study of the complex Monge-Ampère operator on the weighted pluricomplex energy classes. We give more characterizations of the range of the classes Eχ by the complex Monge-Ampère operator. In particular, we prove that a nonnegative Borel measure μ is the Monge-Ampère of a unique function φ∈Eχ if and only if χ(Eχ)⊂L1(dμ). Then we show that if μ=(ddcφ)n for some φ∈Eχ then μ=(ddcu)n for some φ∈Eχ, where f is given boundary data. If moreover the nonnegative Borel measure μ is suitably dominated by the Monge-Ampère capacity, we establish a priori estimates on the capacity of sublevel sets of the solutions. As a consequence, we give a priori bounds of the solution of the Dirichlet problem in the case when the measure has a density in some Orlicz space.

scholarly journals A Priori Bounds for a Class of Elliptic Operators

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Sara Monsurrò ◽  
Maria Salvato ◽  
Maria Transirico
Keyword(s):  
Dirichlet Problem ◽  
Existence Theorem ◽  
Sobolev Spaces ◽  
Differential Operators ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
Unbounded Domains ◽  
Elliptic Operators ◽  
A Priori Bounds ◽  
Uniqueness And Existence

We obtain some a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

scholarly journals Periodic solution for ϕ-Laplacian neutral differential equation

2019 ◽  
Vol 17 (1) ◽  
pp. 172-190 ◽  
Author(s):  
Shaowen Yao ◽  
Zhibo Cheng
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
A Priori ◽  
Neutral Differential Equation ◽  
A Priori Bounds ◽  
T Tau

Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

Sign in / Sign up

Export Citation Format

Share Document