scholarly journals A short proof of the existence of the solution to elliptic boundary problem

2015 ◽  
Vol 3 (3) ◽  
pp. 105
Author(s):  
Alexander G. Ramm
Keyword(s):  
Boundary Problem ◽  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Upper And Lower Solutions ◽  
Parameter Method ◽  
Elliptic Boundary Problem ◽  
Integral Equation Methods ◽  
Priori Estimates ◽  
Uniqueness Of The Solution

<p>There are several methods for proving the existence of the solution to the elliptic boundary problem \(Lu=f \text{in} D,\quad u|_S=0,\quad    (*)\). Here <em>L</em> is an elliptic operator of second order, f is a given function, and uniqueness of the solution to problem (*) is assumed. The known methods for proving the existence of the solution to (*) include variational methods, integral equation methods, method of upper and lower solutions. In this paper a method based on functional analysis is proposed. This method is conceptually simple. It requires some a priori estimates and a continuation in a parameter method, which is well-known.</p>

A difference scheme for equations of ocean dynamics on unstructured grids

2014 ◽  
Vol 29 (3) ◽  
Author(s):  
Alexey V. Drutsa
Keyword(s):  
Difference Scheme ◽  
Existence And Uniqueness ◽  
Large Scale ◽  
A Priori Estimates ◽  
Unstructured Grids ◽  
A Priori ◽  
Ocean Dynamics ◽  
Priori Estimates ◽  
Uniqueness Of The Solution ◽  
Grid Operators

AbstractA difference scheme on unstructured grids is constructed for the system of equations of large scale ocean dynamics. The properties of the grid problem and grid operators are studied, in particular, a series of a priori estimates and the theorem on existence and uniqueness of the solution are proved.

scholarly journals Well-posedness results for the wave equation generated by the Bessel operator

2021 ◽  
Vol 101 (1) ◽  
pp. 11-16
Author(s):  
B. Bekbolat ◽  
◽  
N. Tokmagambetov ◽  
◽  
◽  
...  
Keyword(s):  
Wave Equation ◽  
A Priori Estimates ◽  
A Priori ◽  
Hankel Transform ◽  
Bessel Operator ◽  
Sobolev Type ◽  
Priori Estimates ◽  
Homogeneous Wave ◽  
Uniqueness Of The Solution ◽  
Homogeneous Wave Equation

In this paper, we consider the non-homogeneous wave equation generated by the Bessel operator. We prove the existence and uniqueness of the solution of the non-homogeneous wave equation generated by the Bessel operator. The representation of the solution is given. We obtained a priori estimates in Sobolev type space. This problem was firstly considered in the work of M. Assal [1] in the setting of Bessel-Kingman hypergroups. The technique used in [1] is based on the convolution theorem and related estimates. Here, we use a different approach. We study the problem from the point of the Sobolev spaces. Namely, the conventional Hankel transform and Parseval formula are widely applied by taking into account that between the Hankel transformation and the Bessel differential operator there is a commutation formula [2].

SOLVING STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS BASED ON THE EXPERIMENTAL DATA

2003 ◽  
Vol 13 (03) ◽  
pp. 415-444 ◽  
Author(s):  
IVO BABUŠKA ◽  
KANG-MAN LIU ◽  
RAÚL TEMPONE
Keyword(s):  
Experimental Data ◽  
A Priori Estimates ◽  
Elliptic Boundary ◽  
A Priori ◽  
Independent Random Variables ◽  
Practical Reasons ◽  
Elliptic Boundary Value ◽  
Priori Estimates ◽  
Stochastic Coefficient ◽  
Mutually Independent

We consider a stochastic linear elliptic boundary value problem whose stochastic coefficient a(x, ω) is expressed by a finite number NKL of mutually independent random variables, and transform this problem into a deterministic one. We show how to choose a suitable NKL which should be as low as possible for practical reasons, and we give the a priori estimates for modeling error when a(x, ω) is completely known. When a random function a(x, ω) is selected to fit the experimental data, we address the estimation of the error in this selection due to insufficient experimental data. We present a simple model problem, simulate the experiments, and give the numerical results and error estimates.

An elliptic boundary problem involving a semi-definite weight

2004 ◽  
Vol 134 (1) ◽  
pp. 109-136 ◽  
Author(s):  
M. Faierman
Keyword(s):  
Boundary Conditions ◽  
Weight Function ◽  
Boundary Problem ◽  
Existence And Uniqueness ◽  
Spectral Properties ◽  
Elliptic Boundary ◽  
A Priori ◽  
Operator Pencil ◽  
Bounded Region ◽  
Elliptic Boundary Problem

We consider an elliptic boundary problem defined in a bounded region Ω ⊂ Rn and where the spectral parameter is multiplied by a weight function ω(x). We suppose that ω(x) ≠ 0 for x ∈ Ω, but vanishes in a specified manner on the boundary of Ω. Under limited smoothness assumptions, we derive results pertaining to existence and uniqueness of and a priori estimates for solutions of the boundary problem. If S(λ) denotes the operator pencil induced in L2(Ω) by the boundary problem with zero boundary conditions, then results are also derived pertaining to the spectral properties of S(λ).

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