Volume 11, Issue 2

Let Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}} be a bounded open set. Given 2≤m≤n{2\leq m\leq n}, we construct a convex function u:Ω→ℝ{u\colon\Omega\to\mathbb{R}} whose gradient f=∇⁡u{f=\nabla u} is a Hölder continuous homeomorphism, f is the identity on ∂⁡Ω{\partial\Omega}, the derivative Df has rank m-1{m-1} a.e. in Ω and Df is in the weak Lm{L^{m}} space Lm,w{L^{m,w}}. The proof is based on convex integration and staircase laminates.

We discuss existence of solutions, compactness and stability properties in closed manifolds for the critical Kirchhoff equations (a+b∫M|∇u|2dvg)θ0Δgu+hu=up-1,\Bigg{(}a+b\int_{M}\lvert\nabla u|^{2}\,dv_{g}\Bigg{)}^{\theta_{0}}\Delta_{g}u% +hu=u^{p-1}, where Δg{\Delta_{g}} is the Laplace–Beltrami operator, h is a C1{C^{1}}-function in M , p∈(2,2⋆]{p\in(2,2^{\star}]}, a,b,θ0>0{a,b,\theta_{0}>0} are positive real numbers, and 2⋆{2^{\star}} is the critical Sobolev exponent. A fractional critical dimension d0=2⁢(1+θ0)θ0{d_{0}=\frac{2(1+\theta_{0})}{\theta_{0}}} appears in the critical case p=2⋆{p=2^{\star}}.

In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain ϵ∈ℝ{\epsilon\in\mathbb{R}} and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V , and ϵ. Then we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

In this note, we study the well-posedness of the Cauchy problem for the transport equation in the BMO space and certain Triebel–Lizorkin spaces.

We study the regularity of solutions of one-dimensional variational obstacle problems in W1,1{W^{1,1}} when the Lagrangian is locally Hölder continuous and globally elliptic. In the spirit of the work of Sychev [5, 6, 7], a direct method is presented for investigating such regularity problems with obstacles. This consists of introducing a general subclass ℒ{\mathcal{L}} of W1,1{W^{1,1}}, related in a certain way to one-dimensional variational obstacle problems, such that every function of ℒ{\mathcal{L}} has Tonelli’s partial regularity, and then to prove that, depending on the regularity of the obstacles, solutions of corresponding variational problems belong to ℒ{\mathcal{L}}. As an application of this direct method, we prove that if the obstacles are C1,σ{C^{1,\sigma}}, then every Sobolev solution has Tonelli’s partial regularity.