Closed 𝓐-p Quasiconvexity and Variational Problems with Extended Real-Valued Integrands

2018 ◽  
Vol 24 (4) ◽  
pp. 1605-1624
Author(s):  
Adam Prosinski
Keyword(s):  
Vector Fields ◽  
Additional Assumption ◽  
Integral Functional ◽  
Variational Problems ◽  
Constant Rank ◽  
Differential Constraint ◽  
First Order ◽  
Characteristic Cone ◽  
Lower Semi ◽  
Continuity Assumption

This paper relates the lower semi-continuity of an integral functional in the compensated compactness setting of vector fields satisfying a constant-rank first-order differential constraint, to closed 𝓐-p quasiconvexity of the integrand. The lower semi-continuous envelope of relaxation is identified for continuous, but potentially extended real-valued integrands. We discuss the continuity assumption and show that when it is dropped our notion of quasiconvexity is still equivalent to lower semi-continuity of the integrand under an additional assumption on the characteristic cone of 𝓐.

scholarly journals Off-Shell Noether Currents and Potentials for First-Order General Relativity

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 348
Author(s):  
Merced Montesinos ◽  
Diego Gonzalez ◽  
Rodrigo Romero ◽  
Mariano Celada
Keyword(s):  
General Relativity ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Arbitrary Vector ◽  
Four Dimensions ◽  
First Order ◽  
Direct Form ◽  
Definition Of

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

First-emptiness problems in applied probability under first-order dependent input and general output conditions

1973 ◽  
Vol 10 (02) ◽  
pp. 330-342 ◽  
Author(s):  
J. P. Lehoczky
Keyword(s):  
Markov Chain ◽  
Integral Functional ◽  
Applied Probability ◽  
Input Process ◽  
First Order ◽  
Infinite Reservoir

Results for the first-emptiness time of a semi-infinite reservoir and the integral functional of the process up to first-emptiness time are derived under Markov chain input conditions and general output conditions. The results are further extended to allow an input process which is the sum of k consecutive elements of the Markov chain, k ≧ 1.

Generalizations of Frobenius’ Theorem on Manifolds and Subcartesian Spaces

2007 ◽  
Vol 50 (3) ◽  
pp. 447-459 ◽  
Author(s):  
Jędrzej Śniatycki
Keyword(s):  
Vector Fields ◽  
Integral Manifold ◽  
Constant Rank ◽  
Frobenius Theorem

AbstractLet be a family of vector fields on a manifold or a subcartesian space spanning a distribution D. We prove that an orbit O of is an integral manifold of D if D is involutive on O and it has constant rank on O. This result implies Frobenius’ theorem, and its various generalizations, on manifolds as well as on subcartesian spaces.

A lower semi-continuity result for polyconvex functionals in SBV

2006 ◽  
Vol 136 (2) ◽  
pp. 321-336 ◽  
Author(s):  
Nicola Fusco ◽  
Chiara Leone ◽  
Anna Verde ◽  
Riccardo March
Keyword(s):  
Integral Functional ◽  
Surface Term ◽  
Continuity Theorem ◽  
Continuity Result ◽  
Lower Semi

We prove a semi-continuity theorem for an integral functional made up by a polyconvex energy and a surface term. Our result extends a well-known result by Ball to the BV framework.

scholarly journals Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandro Zagatti
Keyword(s):  
Existence Result ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
First Order ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We study the minimum problem for functionals of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the integrand <inline-formula><tex-math id="M1">\begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document}</tex-math></inline-formula> is not convex in the last variable. We provide an existence result assuming that the lower convex envelope <inline-formula><tex-math id="M2">\begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M4">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is regular and enjoys a special dependence with respect to the i-th single components <inline-formula><tex-math id="M5">\begin{document}$ p_i, q_i, \xi_i $\end{document}</tex-math></inline-formula> of the vector variables <inline-formula><tex-math id="M6">\begin{document}$ p,q,\xi $\end{document}</tex-math></inline-formula>. More precisely, we assume that it is monotone in <inline-formula><tex-math id="M7">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and that it satisfies suitable affinity properties with respect to <inline-formula><tex-math id="M8">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> on the set <inline-formula><tex-math id="M9">\begin{document}$ \{f&gt; \overline{f}\} $\end{document}</tex-math></inline-formula> and with respect to <inline-formula><tex-math id="M10">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.</p>

First Order Partial Functional Differential Equations with Unbounded Delay

2003 ◽  
Vol 10 (3) ◽  
pp. 509-530
Author(s):  
Z. Kamont ◽  
S. Kozieł
Keyword(s):  
Differential Equations ◽  
Integral Functional ◽  
Functional Differential Equations ◽  
Generalized Solutions ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
First Order ◽  
Unbounded Delay ◽  
The Cauchy Problem ◽  
Functional Differential

Abstract The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

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