scholarly journals On the minimization of some nonconvex double obstacle problems

2003 ◽  
Vol 2003 (10) ◽  
pp. 535-551
Author(s):  
A. Elfanni
Keyword(s):  
Numerical Analysis ◽  
Variational Problem ◽  
Minimization Problem ◽  
Lipschitz Functions ◽  
Discrete Space ◽  
Obstacle Problems ◽  
Minimizing Sequences ◽  
Nonconvex Variational Problem ◽  
Admissible Functions

We consider a nonconvex variational problem for which the set of admissible functions consists of all Lipschitz functions located between two fixed obstacles. It turns out that the value of the minimization problem at hand is equal to zero when the obstacles do not touch each other; otherwise, it might be positive. The results are obtained through the construction of suitable minimizing sequences. Interpolating these minimizing sequences in some discrete space, a numerical analysis is then carried out.

Analysis of a Nonconvex Problem Related to Signal Selective Smoothing

1997 ◽  
Vol 07 (03) ◽  
pp. 313-328 ◽  
Author(s):  
M. Chipot ◽  
R. March ◽  
M. Rosati ◽  
G. Vergara Caffarelli
Keyword(s):  
Variational Problem ◽  
Existence Result ◽  
Minimizing Sequences ◽  
Qualitative Properties ◽  
Nonconvex Problem ◽  
Nonconvex Variational Problem ◽  
Selective Smoothing

We study some properties of a nonconvex variational problem. We fail to attain the infimum of the functional that has to be minimized. Instead, minimizing sequences develop gradient oscillations which allow them to reduce the value of the functional. We show an existence result for a perturbed nonconvex version of the problem, and we study the qualitative properties of the corresponding minimizer. The pattern of the gradient oscillations for the original nonperturbed problem is analyzed numerically.

scholarly journals Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Besov Space ◽  
Lipschitz Space ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Quadratic Growth ◽  
Regularity Assumption ◽  
Lipschitz Regularity ◽  
Fixed Boundary ◽  
Higher Differentiability ◽  
Admissible Functions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

Higher differentiability of solutions to a class of obstacle problems under non-standard growth conditions

2019 ◽  
Vol 31 (6) ◽  
pp. 1501-1516 ◽  
Author(s):  
Chiara Gavioli
Keyword(s):  
Variational Inequality ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Differentiability Property ◽  
Integer Order ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Admissible Functions ◽  
Fixed Function

AbstractWe establish the higher differentiability of integer order of solutions to a class of obstacle problems assuming that the gradient of the obstacle possesses an extra integer differentiability property. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form\int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{% for all }\varphi\in\mathcal{K}_{\psi}(\Omega).The main novelty is that the operator {\mathcal{A}} satisfies the so-called {p,q}-growth conditions with p and q linked by the relation\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r},for {r>n}. Here {\psi\in W^{1,p}(\Omega)} is a fixed function, called obstacle, for which we assume {D\psi\in W^{1,2q-p}_{\mathrm{loc}}(\Omega)}, and {\mathcal{K}_{\psi}=\{w\in W^{1,p}(\Omega):w\geq\psi\text{ a.e. in }\Omega\}} is the class of admissible functions. We require for the partial map {x\mapsto\mathcal{A}(x,\xi\/)} a higher differentiability of Sobolev order in the space {W^{1,r}}, with {r>n} satisfying the condition above.

scholarly journals Variational Problem Involving Operator Curl in a Multiconnected Domain

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Junichi Aramaki
Keyword(s):  
Variational Problem ◽  
Minimization Problem ◽  
Vector Fields ◽  
Three Dimensional ◽  
Tangential Component ◽  
Multiconnected Domain

We shall study the problem of minimizing a functional involving the curl of vector fields in a three-dimensional, bounded multiconnected domain with prescribed tangential component on the boundary. The paper is an extension of L2 minimization problem of the curl of vector fields. We shall prove the existence and the estimate of minimizers of more general functional which contains Lp norm of the curl of vector fields.

scholarly journals Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem

2020 ◽  
Vol 13 (4) ◽  
pp. 1269-1290 ◽  
Author(s):  
Annalisa Iuorio ◽  
◽  
Christian Kuehn ◽  
Peter Szmolyan ◽  
Keyword(s):  
Variational Problem ◽  
Numerical Continuation ◽  
Nonconvex Variational Problem

Dynamic Analysis of Flexible Rotors Subjected to Torque and Force

1993 ◽  
Author(s):  
Jong-Seop Yun ◽  
Chong-Won Lee
Keyword(s):  
Numerical Analysis ◽  
Variational Principle ◽  
Dynamic Analysis ◽  
Natural Frequency ◽  
Stability Criterion ◽  
Galerkin’S Method ◽  
Galerkin's Method ◽  
Flexible Rotors ◽  
The Stability ◽  
Admissible Functions

Abstract The effect of the applied direction and magnitude of loads on the stability and natural frequency of flexible rotors is analyzed, when the rotors are subject to nonconservative torque and force. The stability criterion derived from the energy and variational principle is discussed and a general Galerkin’s method which utilizes admissible functions is employed for numerical analysis. Illustrative examples are treated to demonstrate the analytical developments.

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