Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals

1982 ◽  
Author(s):  
Bernard Dacorogna
Keyword(s):  
Lower Semicontinuity ◽  
Weak Continuity ◽  
Linear Functionals ◽  
Weak Lower Semicontinuity ◽  
Non Linear

Weak continuity and weak lower semicontinuity for some compensation operators

1989 ◽  
Vol 113 (3-4) ◽  
pp. 267-279 ◽  
Author(s):  
Pablo Pedregal
Keyword(s):  
Lower Semicontinuity ◽  
Special Class ◽  
Differential Operators ◽  
Necessary Conditions ◽  
Young Measures ◽  
Weak Continuity ◽  
Linear Differential Operators ◽  
Weak Lower Semicontinuity ◽  
Linear Differential ◽  
Quasi Convexity

SynopsisWe study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.

scholarly journals Calculus of variations: A differential form approach

2019 ◽  
Vol 12 (1) ◽  
pp. 57-84 ◽  
Author(s):  
Swarnendu Sil
Keyword(s):  
Continuous Function ◽  
Calculus Of Variations ◽  
Differential Form ◽  
Lower Semicontinuity ◽  
Weak Continuity ◽  
Minimization Problems ◽  
Weak Lower Semicontinuity ◽  
Form Approach

AbstractWe study integrals of the form {\int_{\Omega}f(d\omega_{1},\dots,d\omega_{m})}, where {m\geq 1} is a given integer, {1\leq k_{i}\leq n} are integers, {\omega_{i}} is a {(k_{i}-1)}-form for all {1\leq i\leq m} and {f:\prod_{i=1}^{m}\Lambda^{k_{i}}(\mathbb{R}^{n})\rightarrow\mathbb{R}} is a continuous function. We introduce the appropriate notions of convexity, namely vectorial ext. one convexity, vectorial ext. quasiconvexity and vectorial ext. polyconvexity. We prove weak lower semicontinuity theorems and weak continuity theorems and conclude with applications to minimization problems. These results generalize the corresponding results in both classical vectorial calculus of variations and the calculus of variations for a single differential form.

Measurement of Wiener kernels

1984 ◽  
Vol 16 (1) ◽  
pp. 11-12
Author(s):  
Yoshifusa Ito
Keyword(s):  
Differential Operator ◽  
Mathematical Models ◽  
White Noise ◽  
Linear Systems ◽  
Main Part ◽  
The Other ◽  
Linear Functionals ◽  
Wiener Kernels ◽  
Non Linear ◽  
Non Linear Systems

Since the late 1960s Wiener's theory on the non-linear functionals of white noise has been widely applied to the construction of mathematical models of non-linear systems, especially in the field of biology. For such applications the main part is the measurement of Wiener's kernels, for which two methods have been proposed: one by Wiener himself and the other by Lee and Schetzen. The aim of this paper is to show that there is another method based on Hida's differential operator.

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.

Weak lower semicontinuity for polyconvex integrals in the limit case

2013 ◽  
Vol 51 (1-2) ◽  
pp. 171-193 ◽  
Author(s):  
M. Focardi ◽  
N. Fusco ◽  
C. Leone ◽  
P. Marcellini ◽  
E. Mascolo ◽  
...  
Keyword(s):  
Lower Semicontinuity ◽  
Limit Case ◽  
Weak Lower Semicontinuity

scholarly journals Weak lower semicontinuity for non coercive polyconvex integrals

2008 ◽  
Vol 1 (2) ◽  
Author(s):  
Micol Amar ◽  
Virginia De Cicco ◽  
Paolo Marcellini ◽  
Elvira Mascolo
Keyword(s):  
Lower Semicontinuity ◽  
Weak Lower Semicontinuity
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