Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

In this article we study lower semicontinuous, convex functionals on real Hilbert spaces. In the first part of the article we construct a Banach space that serves as the energy space for such functionals. In the second part we study nonlinear Dirichlet forms, as defined by Cipriani and Grillo, and show, as it is well known in the bilinear case, that the energy space of such forms is a lattice. We define a capacity and introduce the notion quasicontinuity associated with these forms and prove several results, which are well known in the bilinear case.

Global higher integrability for minimisers of convex functionals with (p,q)-growth

We prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands

Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.

A continuously perturbed Dirichlet energy with area-preserving stationary points that ‘buckle’ and occur in equal-energy pairs

We exhibit a family of convex functionals with infinitely many equal-energy $$C^1$$ C 1 stationary points that (i) occur in pairs $$v_{\pm }$$ v ± satisfying $$\det \nabla v_{\pm }=1$$ det ∇ v ± = 1 on the unit ball B in $${\mathbb {R}}^2$$ R 2 and (ii) obey the boundary condition $$v_{\pm }=\text {id}$$ v ± = id on $$ \partial B$$ ∂ B . When the parameter $$\epsilon $$ ϵ upon which the family of functionals depends exceeds $$\sqrt{2}$$ 2 , the stationary points appear to ‘buckle’ near the centre of B and their energies increase monotonically with the amount of buckling to which B is subjected. We also find Lagrange multipliers associated with the maps $$v_{\pm }(x)$$ v ± ( x ) and prove that they are proportional to $$(\epsilon -1/\epsilon )\ln |x|$$ ( ϵ - 1 / ϵ ) ln | x | as $$x \rightarrow 0$$ x → 0 in B. The lowest-energy pairs $$v_{\pm }$$ v ± are energy minimizers within the class of twist maps (see Taheri in Topol Methods Nonlinear Anal 33(1):179–204, 2009 or Sivaloganathan and Spector in Arch Ration Mech Anal 196:363–394, 2010), which, for each $$0\le r\le 1$$ 0 ≤ r ≤ 1 , take the circle $$\{x\in B: \ |x|=r\}$$ { x ∈ B : | x | = r } to itself; a fortiori, all $$v_{\pm }$$ v ± are stationary in the class of $$W^{1,2}(B;{\mathbb {R}}^2)$$ W 1 , 2 ( B ; R 2 ) maps w obeying $$w=\text {id}$$ w = id on $$\partial B$$ ∂ B and $$\det \nabla w=1$$ det ∇ w = 1 in B.