1,721,001 research outputs found
An existence result for a class of non-convex problems of the calculus of variations.
No full textWe consider the functional
\int_\Omega \left[h(\gamma_K(\nabla u(x)))+u(x)\right]dx \qquad u(x)\in W_0^{1,1}(\Omega)
where is the gauge function of a convex set and is a possibly non convex function. In the case is a closed polytope and is a bounded convex set we provide a sufficient condition for the existence of the minimum.
Besides, as a corollary, we give conditions on and that are sufficient to the existence of a minimizer of
\int_\Omega \left[f(\nabla u(x))+u(x)\right]dx \qquad u(x)\in W_0^{1,1}(\Omega)
Local Lipschitz Regularity of Minima For A Scalar Problem of the Calculus of Variations
No full textWe consider a functional I(u) = integral(Omega)f(del(x)) dx on u(0) + W(1,1)(Omega). Under the assumption that f is just convex, we prove a new Comparison Principle, we improve and give a short proof of Cellina's Comparison result for a new class of minimizers. We then extend a local Lipschitz regularity result obtained recently by Clarke for a wider class of functions f and boundary data u(0) satisfying a new one-sided Bounded Slope Condition. A relaxation result follows
A comparison principle and the Lipschitz continuity for minimizers
No full textWe give some conditions that ensure the validity of a Comparison Principle for the Minimizers of integral functionals, without assuming the validity of the Euler-Lagrange equation. We deduce a weak Maximum Principle for (possibly) degenerate elliptic equations and, together with a generalization of the Bounded Slope Condition, a result on the Lipschitz continuity of Minimizers
Lipschitz regularity for minima without strict convexity of the Lagrangian
No full textWe give, in a non-smooth setting, some conditions under which (some of) the minimizers of f(Omega) f(del u(x))dx + g(x,u(x)) dx among the functions in W-1,W-1(Omega) that lie between two Lipschitz functions are Lipschitz. We weaken the usual strict convexity assumption in showing that, if just the faces of the epigraph of a convex function f : R-n --> R are bounded and the boundary datum u(0) satisfies a generalization of the Bounded Slope Condition introduced by A. Cellina then the minima of f Omega f (del u (x)) dx on 1, 1 (Q) whenever they exist, are Lipschitz. A relaxation result follows. u(0) + W-0(1,1) (C) 2007 Elsevier Inc. All rights reserved
On the equivalence of two variational problems
No full textWe consider the following problems
Minimize I(u) =
Ω
f(∇u(x)) + g(x, u(x))dz
on
u ∈ W
1,p
0
(Ω) : u−
(x) ≤ u(x) ≤ u+
(x)
Minimize I(u) =
Ω
f(∇u(x)) + g(x, u(x))dz
on
u ∈ W
1,∞
0
(Ω) : ∇u(x) ∈ K
where f : R
n → R is a convex function, Ω is an open bounded subset of
R, K is a closed convex subset of R
n
such that 0 ∈ int K and u−
and u+
are suitable obstacles. We give conditions on the function g under which the
two problems are equivalen
Holder regularity for a classical problem of the calculus of variations
No full textLet Omega subset of R(n) be bounded, open and convex. Let F : R(n) -> R be convex, coercive of order p > 1 and such that the diameters of the projections of the faces of the epigraph of F are uniformly bounded. Then every minimizer of integral(Omega) F(del v(x))dx, v is an element of phi + W(0)(1,1)(Omega, R), is Holder continuous in (Omega) over bar of order p-1/n+p-1 whenever phi is Lipschitz on partial derivative Omega. A similar result for non convex Lagrangians that admit a minimizer follows
Non-Occurrence of a Gap Between Bounded and Sobolev Functions for a Class of Nonconvex Lagrangians
Full text linkWe consider the classical functional of the Calculus of Variations of the form I(u) = ZΩ F(x, u(x), ∇u(x)) dx where Ω is a bounded open subset of Rn and F : Ω×R×Rn → R is a given Carathéodory function; the admissible functions u coincide with a given Lipschitz function on ∂Ω. We formulate some conditions under which a given function in φ + W01,p(Ω) with I(u) < +∞ can be approximated by a sequence of functions uk ∈ φ+W01,p(Ω)∩L∞ converging to u in the norm of W1,p, and such that I(uk) → I(u). The problem is strictly related with the non occurrence of the Lavrentiev gap
Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
Full text linkWe consider the classical functional of the Calculus of Variations of the form
I(u)=∫ΩF(x,u(x),∇u(x))dx,
where Ω is a bounded open subset of Rn and F : Ω × R × Rn → R is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function φ on ∂Ω. We formulate some conditions under which a given function in φ + W1,p0(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with φ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth
Quantitative analysis of the Hopf bifurcation in the Goodwin -dimensional metabolic control system. 29 (1991), no. 8, 733--742.
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