Relaxation of One-Dimensional Variational Problems

2000 ◽  
pp. 3-33
Author(s):  
Andrej Cherkaev
Keyword(s):  
Variational Problems ◽  
One Dimensional

scholarly journals Cosmic Analogues of Classic Variational Problems

Universe ◽  
2020 ◽  
Vol 6 (6) ◽  
pp. 71 ◽  
Author(s):  
Valerio Faraoni
Keyword(s):  
Relativistic Particle ◽  
Friedmann Equation ◽  
Variational Calculus ◽  
Variational Problems ◽  
Particle Mechanics ◽  
One Dimensional ◽  
Spatially Homogeneous

Several classic one-dimensional problems of variational calculus originating in non-relativistic particle mechanics have solutions that are analogues of spatially homogeneous and isotropic universes. They are ruled by an equation which is formally a Friedmann equation for a suitable cosmic fluid. These problems are revisited and their cosmic analogues are pointed out. Some correspond to the main solutions of cosmology, while others are analogous to exotic cosmologies with phantom fluids and finite future singularities.

scholarly journals Generic uniqueness of a minimal solution for variational problems on a torus

2002 ◽  
Vol 7 (3) ◽  
pp. 143-154
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Rotation Number ◽  
Rational Number ◽  
Variational Problems ◽  
Minimal Solution ◽  
One Dimensional ◽  
Generic Uniqueness

We study minimal solutions for one-dimensional variational problems on a torus. We show that, for a generic integrand and any rational numberα, there exists a unique (up to translations) periodic minimal solution with rotation numberα.

scholarly journals Regularity of Weak Solution of Variational Problems Modeling the Cosserat Micropolar Elasticity

2020 ◽  
Author(s):  
Yimei Li ◽  
Changyou Wang
Keyword(s):  
Weak Solution ◽  
Harmonic Maps ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Energy Functional ◽  
Geometrically Nonlinear ◽  
Micropolar Elasticity ◽  
One Dimensional ◽  
The Poisson Equation ◽  
System Coupling

Abstract In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solution is stationary, then its singular set is discrete for $2<p<3$ and has zero one-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.

Remarks on necessary conditions for minimizers of one-dimensional variational problems

2002 ◽  
Vol 48 (7) ◽  
pp. 979-993 ◽  
Author(s):  
C. Marcelli ◽  
E. Outkine ◽  
M. Sytchev
Keyword(s):  
Necessary Conditions ◽  
Variational Problems ◽  
One Dimensional

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>

scholarly journals Pathological solutions to the Euler–Lagrange equation and existence/regularity of minimizers in one-dimensional variational problems

2017 ◽  
Vol 355 (3) ◽  
pp. 359-362 ◽  
Author(s):  
Richard Gratwick ◽  
Aidys Sedipkov ◽  
Mikhail Sychev ◽  
Aris Tersenov
Keyword(s):  
Lagrange Equation ◽  
Variational Problems ◽  
One Dimensional ◽  
Regularity Of Minimizers
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