Embeddings for A\mathbb{A}-weakly differentiable functions on domains

We prove that the critical embedding $\mathrm{W}^{\mathbb{A},1}(B)\hookrightarrow \mathrm{W}^{k-1,\frac{n}{n-1}}$ holds if and only if the $k$-homogeneous, linear differential operator $\mathbb{A}$ on $\mathbb{R}^n$ from $\mathbb{R}^N$ to $\mathbb{R}^m$ has finite dimensional null-space. Here $B$ is a ball in $\mathbb{R}^n$ and $\mathrm{W}^{\mathbb{A},1}(B)$ denotes the space of maps $u\in \mathrm{L}^1(B,\mathbb{R}^N)$ such that the vector valued distribution $\mathbb{A}u$ is an integrable map. The result was previously known only for several examples of $\mathbb{A}$. Our result contrasts the homogeneous embedding in full-space. Namely, Van Schaftingen proved that $\dot{\mathrm{W}}{^{\mathbb{A},1}}(\mathbb{R}^n)\hookrightarrow \dot{\mathrm{W}}{^{k-1,\frac{n}{n-1}}}$ if and only if $\mathbb{A}$ is elliptic and cancelling. We show that this condition is (strictly) implied by $\mathbb{A}$ having finite dimensional null-space.Comment: 23 pages, 1 tabl

Electro-rheological fluids under random influences: martingale and strong solutions

We study generalised Navier--Stokes equations governing the motion of an electro-rheological fluid subject to stochastic perturbation. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise and (iii) a random character of the variable exponent $p=p(\omega,t,x)$ (as a result of a random electric field). We show the existence of a weak martingale solution provided the variable exponent satisfies $p\geq p^->\frac{3n}{n+2}$ ($p^->1$ in two dimensions). Under additional assumptions we obtain also pathwise solutions

On the Trace Operator for Functions of Bounded A\mathbb{A}-Variation

In this paper, we consider the space $\mathrm{BV}^{\mathbb A}(\Omega)$ of functions of bounded $\mathbb A$-variation. For a given first order linear homogeneous differential operator with constant coefficients $\mathbb A$, this is the space of $L^1$--functions $u:\Omega\rightarrow\mathbb R^N$ such that the distributional differential expression $\mathbb A u$ is a finite (vectorial) Radon measure. We show that for Lipschitz domains $\Omega\subset\mathbb R^{n}$, $\mathrm{BV}^{\mathbb A}(\Omega)$-functions have an $L^1(\partial\Omega)$-trace if and only if $\mathbb A$ is $\mathbb C$-elliptic (or, equivalently, if the kernel of $\mathbb A$ is finite dimensional). The existence of an $L^1(\partial\Omega)$-trace was previously only known for the special cases that $\mathbb A u$ coincides either with the full or the symmetric gradient of the function $u$ (and hence covered the special cases $\mathrm{BV}$ or $\mathrm{BD}$). As a main novelty, we do not use the fundamental theorem of calculus to construct the trace operator (an approach which is only available in the $\mathrm{BV}$- and $\mathrm{BD}$-setting) but rather compare projections onto the nullspace as we approach the boundary. As a sample application, we study the Dirichlet problem for quasiconvex variational functionals with linear growth depending on $\mathbb A u$

Partial Regularity for BV Minimizers

We establish an $\varepsilon$-regularity result for the derivative of a map of bounded variation that minimizes a strongly quasiconvex variational integral of linear growth, and, as a consequence, the partial regularity of such BV minimizers. This result extends the regularity theory for minimizers of quasiconvex integrals on Sobolev spaces to the context of maps of bounded variation. Previous partial regularity results for BV minimizers in the linear growth set-up were confined to the convex situation