Differentiability and higher integrability results for local minimizers of splitting-type variational integrals in 2D with applications to nonlinear Hencky-materials

Bildhauer, Michael; Fuchs, Martin
January 2010
Calculus of Variations & Partial Differential Equations;Jan2010, Vol. 37 Issue 1/2, p167
Academic Journal
We prove higher integrability and differentiability results for local minimizers u: $${\mathbb {R}^2\supset\Omega\to\mathbb {R}^M}$$, M = 1, of the splitting-type energy $${\int_{\Omega}[h_1(|\partial_1 u|)+h_2(|\partial_2 u|)]\,{\rm d}x}$$ . Here h1, h2 are rather general N-functions and no relation between h1 and h2 is required. The methods also apply to local minimizers u: $${\mathbb {R}^2\supset\Omega \to \mathbb {R}^2}$$ of the functional $${\int_{\Omega}[h_1(|{\rm div}\,{\rm u}|)+h_2(|\varepsilon^D(u)|)]\,{\rm d}x}$$ so that we can include some variants of so-called nonlinear Hencky-materials. Further extensions concern non-autonomous problems.


Related Articles

  • A reverse log-Sobolev inequality in the Segal-Bargmann space. Sontz, Stephen Bruce // Journal of Mathematical Physics;Mar1999, Vol. 40 Issue 3, p1677 

    Reports on the use of results on the properties of the reproducing kernel of the Segal-Bargmann space to demonstrate a family of energy-entropy inequalities or the log-Sobolev inequality. Possible relation of the Segal-Bargmann transformation with reverse hypercontractivity; Notations and...

  • A Density Problem for Sobolev Spaces on Planar Domains. Koskela, Pekka; Zhang, Yi // Archive for Rational Mechanics & Analysis;Oct2016, Vol. 222 Issue 1, p1 

    We prove that for a bounded, simply connected domain $${\Omega \subset {\mathbb{R}^{2}}}$$ , the Sobolev space $${W^{1,\,\infty}(\Omega)}$$ is dense in $${W^{1,\,p}(\Omega)}$$ for any $${1\leqq p < \infty}$$ . Moreover, we show that if $${\Omega}$$ is Jordan, then...

  • Weak convergence theory for strong materials with p( x)-growth. Sychev, M. // Doklady Mathematics;May2013, Vol. 87 Issue 3, p334 

    Let L: Ω × R × R → R be a Caratheodory integrand with Under these assumptions the weak convergence theory holds for the integral functional $J(u): = \int\limits_\Omega {L(x,u(x),Du(x))dx} $ without further requirements. Weak convergence theory includes lower seraicontinuity...

  • On Local Hörmander-Beurling Spaces. Villegas, Jairo // Turkish Journal of Mathematics;2004, Vol. 28 Issue 4, p387 

    In this paper we aim to extend a result of Hörmander's, that Bp ,kloc (Ω) ⊂ Cm (Ω) if (1+|⋅|)m/k ∈ Lp′, to the setting of vector valued local Hörmander-Beurling spaces, as well as to show that the space ∩j=1∞ Bpj, kjloc (Ω, E) (1 ≤ pj...

  • Regularity Results for the Generalized Beltrami System. Shen Zhou Zheng, Mümün // Acta Mathematica Sinica;Mar2004, Vol. 20 Issue 2, p293 

    For the generalized Beltrami system with two characteristic matrices, we deal with the regularity of its very weak solutions in the Sobolev class Wloc1,r(Ω Rn) (1 < r < n). By changing the generalized Beltrami system into a class of a divergent...

  • The dichotomy between traces on d-sets Γ in ℝ n and the density of D(ℝ n \Γ) in function spaces. Triebel, Hans // Acta Mathematica Sinica;Apr2008, Vol. 24 Issue 4, p539 

    A space A (ℝ n ) with A = B or A = F and s ∈ ℝ, 0 < p,q < ∞ either has a trace in L p (Γ), where Γ is a compact d-set in ℝ n with 0 < d < n, or D(ℝ n Γ) is dense in it. Related dichotomy numbers are introduced and calculated.

  • A new approach to Sobolev spaces and connections to $\mathbf\Gamma$ -convergence. Augusto C. Ponce // Calculus of Variations & Partial Differential Equations;Mar2004, Vol. 19 Issue 3, p229 

    This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit $$\int_\Omega \! \int_\Omega \omega\left( \frac{|f(x)-f(y)|}{|x-y|} \right) \rho_\varepsilon(x-y) \, dx \, dy \quad \text{as $\varepsilon \downarrow 0$}, $$ for $\omega : [0,\infty) \to...

  • Error analysis of spectral method on a triangle. Li-Lian Wang // Advances in Computational Mathematics;May2007, Vol. 26 Issue 4, p473 

    Abstract??In this paper, the orthogonal polynomial approximation on triangle, proposed by Dubiner, is studied. Some approximation results are established in certain non-uniformly Jacobi-weighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral...

  • On an Inverse Problem for a Parabolic Equation. Tkachenko, D. S. // Mathematical Notes;May/Jun2004, Vol. 75 Issue 5/6, p676 

    In this paper, we study the inverse problem of the reconstruction of the right-hand side of special form for a parabolic equation in u in which the coefficients of u_t and u depend on (x,t), with overdetermination given by the integral of the...


Read the Article


Sign out of this library

Other Topics