Butterfly support for off diagonal coefficients and boundedness of solutions to quasilinear elliptic systems

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

Hölder regularity for porous medium systems

We establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. $$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$ ∂ t u - div ( m | u | m - 1 D u ) = 0 , m > 0 . As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

Competing Transport Futures: Tensions between Imaginaries of Electrification and Biogas Fuel in Sweden

The choice of fuels has frequently been at the center of debates about how a future low-carbon mobility system can be achieved. This paper introduces two visions of biogas fuels and electricity using material from interviews and documents in Swedish transport. These visions are analyzed as interrelated sociotechnical imaginaries. To better understand the way visions of biogas and electric vehicles (EVs) dynamically shape and condition each other, four dimensions of sociotechnical imaginaries are further developed: spatial boundedness, temporality, coherence and contestation, and the socio-material relations they are associated with. Imaginaries of biogas and EVs differ with respect to these characteristics. The biogas imaginary is made up of locally bounded visions of the desirable future, showing how imaginaries can be fragmented and contested, often because of their embeddedness in local socio-material systems of resource use. This local boundedness is exemplified by contrasting cases of contested biogas imaginaries in the Swedish municipalities of Linköping and Malmö. The imaginary of EVs, in contrast, is more uniform nationally and even influenced by international expectations that in the future vehicles will be shared, electric, and autonomous. The qualities of these imaginaries shape the way they interrelate and coevolve as sociotechnical changes of the transport system unfold.

Regularity results for a class of obstacle problems with p, q−growth conditions

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ΩF(x, Dz) : z ∈ 𝛫ψ(Ω)}. Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝn, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.

Growth conditions and regularity, an optimal local boundedness result

We prove local boundedness of local minimizers of scalar integral functionals [Formula: see text], [Formula: see text] where the integrand satisfies [Formula: see text]-growth of the form [Formula: see text] under the optimal relation [Formula: see text].