Search

A way to measure the lower growth rate of φ:Ω×[0,∞)→[0,∞)$$ \varphi :\Omega \times \left[0,\infty \right)\to \left[0,\infty \right) $$ is to require t↦φ(x,t)t−r$$ t\mapsto \varphi \left(x,t\right){t}^{-r} $$ to be increasing in (0,∞)$$ \left(0,\infty \right) $$. If this condition holds with r=1$$ r=1 $$, then infu∈f+W01,φ(Ω)∫Ωφ(x,|∇u|)dx$$ \underset{u\in f+{W}_0^{1,\varphi}\left(\Omega \right)}{\operatorname{inf}}{\int}_{\Omega}\varphi \left(x,|\nabla u|\right)\kern0.1em dx $$with boundary values f∈W1,φ(Ω)$$ f\in {W}^{1,\varphi}\left(\Omega \right) $$ does not necessarily have a minimizer. However, if φ$$ \varphi $$ is replaced by φp$$ {\varphi}^p $$, then the growth condition holds with r=p>1$$ r=p>1 $$ and thus (under some additional conditions) the corresponding energy integral has a minimizer. We show that a sequence (up)$$ \left({u}_p\right) $$ of such minimizers converges when p→1+$$ p\to {1}^{+} $$ in a suitable BV$$ \mathrm{BV} $$‐type space involving generalized Orlicz growth and obtain the Γ$$ \Gamma $$‐convergence of functionals with fixed boundary values and of functionals with fidelity terms. [ABSTRACT FROM AUTHOR]

We propose a homogenized supremal functional rigorously derived via L p -approximation by functionals of the type ess-sup x ∈ Ω f (x , D u) , when Ω is a bounded open set of ℝ n and u ∈ W 1 , ∞ (Ω ; ℝ d). The homogenized functional is also deduced directly in the case where the sublevel sets of f (x , ⋅) satisfy suitable convexity properties, as a corollary of homogenization results dealing with pointwise gradient constrained integral functionals. [ABSTRACT FROM AUTHOR]

Integrals of the Calculus of Variations with p , q {p,q} -growth may have not smooth minimizers, not even bounded, for general p , q {p,q} exponents. In this paper we consider the scalar case, which contrary to the vector-valued one allows us not to impose structure conditions on the integrand f ⁢ (x , ξ) {f(x,\xi)} with dependence on the modulus of the gradient, i.e. f ⁢ (x , ξ) = g ⁢ (x , | ξ |) {f(x,\xi)=g(x,|\xi|)}. Without imposing structure conditions, we prove that if q p {\frac{q}{p}} is sufficiently close to 1, then every minimizer is locally Lipschitz-continuous. [ABSTRACT FROM AUTHOR]

We are interested in the regularity of local minimizers of energy integrals of the Calculus of Variations. Precisely, let Ω be an open subset of Rn. Let f (x, ξ) be a real function defined in Ω X Rn satisfying the growth condition |fξx (x,ξ)| ≤ h (x) |ξ|p-1, for x ∊ Ω and ξ ∊ Rn with |ξ| ≥ M0 for some M0 ≥ 0, with h ∊ Lrloc (Ω) for some r > n. This growth condition is more general than those considered in the mathematical literature and allows us t o handle some cases recently studied in similar contexts. We associate to f (x, ξ) the so-called natural p--growth conditions on the second derivatives fξξ (x, ξ); i.e., (p -- 2) --growth for |fξξ (x, ξ)| from above and (p -- 2) --growth from below for the quadratic form (fξξ (x, ξ) λ, λ); for details see either (3) or (7) below. We prove that these conditions are sufficient for the local Lipschitz continuity of any minimizer u ∊ W1,ploc (Ω) of the energy integral fΩf(x, Du (x)) dx. [ABSTRACT FROM AUTHOR]

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form F (v , Ω) = ∫ Ω F (x , D v (x)) d x , where, for n ≥ 2 and N ≥ 1 , Ω is a bounded open set in R n , u ∈ W 1 , 1 (Ω , R N) and the energy density F : Ω × R N × n → R satisfies the so called variable growth conditions. The main novelty here is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the x variable. [ABSTRACT FROM AUTHOR]