scholarly journals Globally Lipschitz minimizers for variational problems with linear growth

2018 ◽  
Vol 24 (4) ◽  
pp. 1395-1413 ◽  
Author(s):  
Lisa Beck ◽  
Miroslav Bulíček ◽  
Erika Maringová
Keyword(s):  
Linear Growth ◽  
Existence Of Solutions ◽  
Direct Method ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Boundary Values ◽  
Equivalent Formulation ◽  
Radial Structure ◽  
Variational Integrals ◽  
Necessary And Sufficient

We study the minimization of convex, variational integrals of linear growth among all functions in the Sobolev space W1,1 with prescribed boundary values (or its equivalent formulation as a boundary value problem for a degenerately elliptic Euler–Lagrange equation). Due to insufficient compactness properties of these Dirichlet classes, the existence of solutions does not follow in a standard way by the direct method in the calculus of variations and in fact might fail, as it is well-known already for the non-parametric minimal surface problem. Assuming radial structure, we establish a necessary and sufficient condition on the integrand such that the Dirichlet problem is in general solvable, in the sense that a Lipschitz solution exists for any regular domain and all prescribed regular boundary values, via the construction of appropriate barrier functions in the tradition of Serrin’s paper [J. Serrin, Philos. Trans. R. Soc. Lond., Ser. A 264 (1969) 413–496].

A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems

1994 ◽  
Vol 124 (3) ◽  
pp. 437-471 ◽  
Author(s):  
Gero Friesecke
Keyword(s):  
Solid Phase ◽  
Young Measure ◽  
Variational Problems ◽  
Boundary Values ◽  
Necessary And Sufficient Condition ◽  
Linear Deformation ◽  
Almost Everywhere ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Unique Limit

For scalar variational problemssubject to linear boundary values, we determine completely those integrandsW: ℝn→ ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.As a corollary, we show that in case of nonattainment (and providedWgrows superlinearly at infinity), every minimising sequence converges weakly but not strongly inW1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.Connections with solid–solid phase transformations are indicated.

Existence of Solutions of an Ill-Posed Problem for the Vibrating String

1982 ◽  
Vol 25 (1) ◽  
pp. 29-36
Author(s):  
L. L. Campbell
Keyword(s):  
Dirichlet Problem ◽  
Simple Formula ◽  
Existence Of Solutions ◽  
Sufficient Condition ◽  
String Equation ◽  
Boundary Values ◽  
Necessary And Sufficient Condition ◽  
Vibrating String ◽  
Ill Posed ◽  
Necessary And Sufficient

AbstractThe Dirichlet problem is examined for the vibrating string equation on a rectangle with commensurable sides. As is well-known, a solution, if it exists, is not unique. A necessary and sufficient condition is obtained on the boundary values for existence of solutions. A simple formula for the solution is obtained.

scholarly journals A necessary and sufficient condition for the continuity of local minima of parabolic variational integrals with linear growth

2017 ◽  
Vol 10 (3) ◽  
pp. 209-221
Author(s):  
Emmanuele DiBenedetto ◽  
Ugo Gianazza ◽  
Colin Klaus
Keyword(s):  
Linear Growth ◽  
Continuity Condition ◽  
Sufficient Condition ◽  
Local Minima ◽  
Necessary And Sufficient Condition ◽  
Fast Decay ◽  
Laplacian Equation ◽  
Variational Integrals ◽  
Necessary And Sufficient ◽  
Proper Solutions

AbstractFor proper minimizers of parabolic variational integrals with linear growth with respect to {|Du|}, we establish a necessary and sufficient condition for u to be continuous at a point {(x_{o},t_{o})}, in terms of a sufficient fast decay of the total variation of u about {(x_{o},t_{o})}. These minimizers arise also as proper solutions to the parabolic 1-Laplacian equation. Hence, the continuity condition continues to hold for such solutions.

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Jianke Zhang ◽  
Gaofeng Wang ◽  
Xiaobin Zhi ◽  
Chang Zhou
Keyword(s):  
Fractional Derivative ◽  
Lagrange Function ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Lagrange Equations ◽  
Hukuhara Difference ◽  
Generalized Hukuhara Difference ◽  
Fractional Variational Problems ◽  
Fractional Calculus Of Variations ◽  
Necessary And Sufficient

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

scholarly journals An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

2020 ◽  
Vol 8 (2) ◽  
pp. 590-601
Author(s):  
Melani Barrios ◽  
Gabriela Reyero
Keyword(s):  
Exact Solution ◽  
Variational Problem ◽  
Lagrange Equation ◽  
Integral Form ◽  
Variational Problems ◽  
The Other ◽  
Isoperimetric Problems ◽  
Caputo Derivatives ◽  
Fractional Variational Problems ◽  
Derivatives Of

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

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