Functional Analysis and Sobolev Spaces

2013 ◽  
pp. 1-26
Author(s):  
Fioralba Cakoni ◽  
David Colton
Keyword(s):  
Functional Analysis ◽  
Sobolev Spaces

scholarly journals Existence of Weak Solutions of Linear Subelliptic Dirichlet Problems with Rough Coefficients

2012 ◽  
Vol 64 (6) ◽  
pp. 1395-1414 ◽  
Author(s):  
Scott Rodney
Keyword(s):  
Functional Analysis ◽  
Quadratic Form ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Principal Part ◽  
Regularity Theory ◽  
Existence Theory ◽  
Dirichlet Problems ◽  
Existence Of Weak Solutions ◽  
Rough Coefficients

Abstract This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the formThe principal part ξ'P(x)ξ of the above equation is assumed to be comparable to a quadratic form Q(x,ξ)=ξ'Q(x)ξ that may vanish for non-zero ξ ∊ ℝn. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces QH1 (Θ)=W1,2(Θ,Q) and QH10(Θ)= W1,20 (Θ,Q)as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

15.—Differentiability on the Boundary of the Solutions of Linear Elliptic Boundary Value Problems

1975 ◽  
Vol 73 ◽  
pp. 235-249 ◽  
Author(s):  
Renate Schappel
Keyword(s):  
Functional Analysis ◽  
Boundary Value Problems ◽  
Sobolev Spaces ◽  
Weak Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Elliptic Boundary Value Problems ◽  
Elliptic Boundary Value ◽  
Regularity Of Weak Solutions ◽  
Linear Elliptic Boundary

SynopsisThe present paper is concerned with the problem of regularity of weak solutions of boundary value problems. We shall present a new method to prove differentiability on the boundary. This method was developed in our thesis [12] within the theory of abstract Sobolev spaces, introduced by Stummel [14]. Here, we shall describe it by applying it to elliptic boundary value problems. It will be seen that the advantage of this method consists in the fact that it is based on functional analysis only and therefore may be used for other types of differential equations as well.

scholarly journals Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ge Dong ◽  
Xiaochun Fang
Keyword(s):  
Functional Analysis ◽  
Sobolev Spaces ◽  
Linear Functional ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Solution Set ◽  
Analysis Method ◽  
Boundary Equation ◽  
Elliptic Differential Equations ◽  
Carathéodory Functions

In this paper, we study the solution set of the following Dirichlet boundary equation: − div a 1 x , u , D u + a 0 x , u = f x , u , D u in Musielak-Orlicz-Sobolev spaces, where a 1 : Ω × ℝ × ℝ N ⟶ ℝ N , a 0 : Ω × ℝ ⟶ ℝ , and f : Ω × ℝ × ℝ N ⟶ ℝ are all Carathéodory functions. Both a 1 and f depend on the solution u and its gradient D u . By using a linear functional analysis method, we provide sufficient conditions which ensure that the solution set of the equation is nonempty, and it possesses a greatest element and a smallest element with respect to the ordering “≤,” which are called barrier solutions.

Case Formulation and Design of Behavioral Treatment Programs

2003 ◽  
Vol 19 (3) ◽  
pp. 164-174 ◽  
Author(s):  
Stephen N. Haynes ◽  
Andrew E. Williams
Keyword(s):  
Functional Analysis ◽  
Behavior Problems ◽  
Behavioral Treatment ◽  
Clinical Case ◽  
Relative Magnitude ◽  
Mechanisms Of Action ◽  
Treatment Programs ◽  
Case Formulation ◽  
Functional Relations ◽  
Causal Variables

Summary: We review the rationale for behavioral clinical case formulations and emphasize the role of the functional analysis in the design of individualized treatments. Standardized treatments may not be optimally effective for clients who have multiple behavior problems. These problems can affect each other in complex ways and each behavior problem can be influenced by multiple, interacting causal variables. The mechanisms of action of standardized treatments may not always address the most important causal variables for a client's behavior problems. The functional analysis integrates judgments about the client's behavior problems, important causal variables, and functional relations among variables. The functional analysis aids treatment decisions by helping the clinician estimate the relative magnitude of effect of each causal variable on the client's behavior problems, so that the most effective treatments can be selected. The parameters of, and issues associated with, a functional analysis and Functional Analytic Clinical Case Models (FACCM) are illustrated with a clinical case. The task of selecting the best treatment for a client is complicated because treatments differ in their level of specificity and have unequally weighted mechanisms of action. Further, a treatment's mechanism of action is often unknown.

The Social Order Yields to Functional Analysis

1958 ◽  
Vol 3 (6) ◽  
pp. 158-160
Author(s):  
LAWRENCE SCHLESINGER
Keyword(s):  
Functional Analysis ◽  
Social Order ◽  
The Social
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