In this paper, we study a connection between disintegration of measures and geometric properties of probability spaces. We prove a disintegration theorem, addressing disintegration from the perspective of an optimal transport problem. We look at the disintegration of transport plans, which are used to define and study disintegration maps. Using these objects, we study the regularity and absolute continuity of disintegration of measures. In particular, we exhibit conditions for which the disintegration map is weakly continuous and one can obtain a path of measures given by this map. We show a rigidity condition for the disintegration of measures to be given into absolutely continuous measures.

We derive a global higher regularity result for weak solutions of the linear relaxed micromorphic model on smooth domains. The governing equations consist of a linear elliptic system of partial differential equations that is coupled with a system of Maxwell-type. The result is obtained by combining a Helmholtz decomposition argument with regularity results for linear elliptic systems and the classical embedding of $H(\operatorname {div};\Omega )\cap H_0(\operatorname {curl};\Omega )$ into $H^1(\Omega )$.

Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$, $d\geq 2$, equipped with surface measure $\sigma _{d-1}$. An instance of our main result concerns the regularity of solutions of the convolution equation

$$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$

$a\in C^\infty (\mathbb {S}^{d-1})$

$q\geq 2(d+1)/(d-1)$

$f\in L^2(\mathbb {S}^{d-1})$

$C^\infty (\mathbb {S}^{d-1})$

$\mathbb {S}^{d-1}$

$C^\infty $

Let Ω ⊂ ℝn be a bounded Lipschitz domain. Let $L: {\mathbb R}^n\rightarrow \bar {\mathbb R}= {\mathbb R}\cup \{+\infty \}$ be a continuous function with superlinear growth at infinity, and consider the functional $\mathcal {I}(u)=\int \nolimits _\Omega L(Du)$, u ∈ W1,1(Ω). We provide necessary and sufficient conditions on L under which, for all f ∈ W1,1(Ω) such that $\mathcal {I}(f) < +\infty $, the problem of minimizing $\mathcal {I}(u)$ with the boundary condition u|∂Ω = f has a solution which is stable, or – alternatively – is such that all of its solutions are stable. By stability of $\mathcal {I}$ at u we mean that $u_k\rightharpoonup u$ weakly in W1,1(Ω) together with $\mathcal {I}(u_k)\to \mathcal {I}(u)$ imply uk → u strongly in W1,1(Ω). This extends to general boundary data some results obtained by Cellina and Cellina and Zagatti. Furthermore, with respect to the preceding literature on existence results for scalar variational problems, we drop the assumption that the relaxed functional admits a continuous minimizer.

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine, in particular, the relationship between the positivity of the Jacobian det ∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit explicit twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma }: \Omega \to {\open R}^2$ in a model, two-dimensional case. We exploit the Jacobian constraint $\det \nabla u_{\sigma} \gt 0$ a.e. to obtain regularity results that apply ‘up to the boundary’ of domains with corners. It is shown that the unique shear map minimizer has the properties that (i) $\det \nabla u_{\sigma }$ is strictly positive on one part of the domain Ω, (ii) $\det \nabla u_{\sigma } = 0$ necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma }$ is not continuous on the whole domain.

We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be singular at the boundary of a domain and discuss implications for practical applications. We furthermore give partial regularity results for the phase-fields uε at the boundary in terms of boundary conditions and counterexamples without boundary conditions.

Let $\Omega \subset {{\mathbb{R}}^{n}}$ be a bounded Lipschitz domain and consider the energy functional

1

$$F[u,\Omega ]\,:=\,\,{{\int }_{\Omega }}\text{F}(\triangledown u(x))\,d\text{x,}$$

over the space of ${{W}^{1,2}}(\Omega ,{{\mathbb{R}}^{m}})$ where the integrand $\text{F}:{{\mathbb{M}}_{m\times n}}\to \mathbb{R}$ is a smooth uniformly convex function with bounded second derivatives. In this paper we address the question of regularity for solutions of the corresponding system of Euler–Lagrange equations. In particular, we introduce a class of singularmaps referred to as traceless and examine themas a new counterexample to the regularity of minimizers of the energy functional $F[\cdot ,\Omega ]$ using a method based on null Lagrangians.

The squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.