We establish the higher differentiability of the local minimizers to a class of non autonomous convex integral functionals satisfying anisotropic subquadratic growth conditions, that include, as a particular case, those with orthotropic structure. The result is obtained under a gap bound on the exponents p_i, that guarantees the local boundedness of the minimizers and under a suitable Sobolev assumption on the map that measures the oscillation of the energy density with respect to the x variable, that is independent on the dimension.

We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the H & ouml;lder regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.

We prove a sharp higher differentiability result for local minimizers of functionals of the form [Formula: see text] with non-autonomous integrand [Formula: see text] which is convex with respect to the gradient variable, under [Formula: see text]-growth conditions, with [Formula: see text]. The main novelty here is that the results are obtained assuming that the partial map [Formula: see text] has weak derivatives in some Lebesgue space [Formula: see text] and the datum [Formula: see text] is assumed to belong to a suitable Lebesgue space [Formula: see text]. We also prove that it is possible to weaken the assumption on the datum [Formula: see text] and on the map [Formula: see text], if the minimizers are assumed to be a priori bounded.

We prove existence of solutions to a nonlinear transport equation in the plane, for which the velocity field is obtained as the convolution of the classical Cauchy kernel with the unknown. Even though the initial datum is bounded and compactly supported, the velocity field may have unbounded divergence. The proof is based on the compactness property of quasiconformal mappings.

Among those nearly incompressible vector fields v:Rn→Rn with |x|log⁡|x| growth at infinity, we give a pointwise characterization of the ones for which curlv=Dv−Dtv belongs to L∞. When n=2 we can go further and describe, still in pointwise terms, the vector fields v:R2→R2 for which |divv|+|curlv|∈L∞.

We obtain sharp rotation bounds for the subclass of homeomorphisms f:ℂ→ℂ of finite distortion which have distortion function in L^p_loc , p>1 , and for which a Hölder continuous inverse is available. The interest in this class is partially motivated by examples arising from fluid mechanics. Our rotation bounds hereby presented improve the existing ones, for which the Hölder continuity is not assumed. We also present examples proving sharpness.

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of Hölder regularity, higher than what is given by the classical exponent 1/K.

We prove the local Lipschitz continuity and the higher differentiability of local minimizers of integral functionals with non autonomous integrand which is degenerate convex with respect to the gradient variable. The main novelty here is that the results are obtained assuming that the coefficients have weak derivative in an almost critical Zygmund class and the datum f is assumed to belong to the same Zygmund class.

We prove well-posedness of linear scalar conservation laws using only assumptions on the growth and the modulus of continuity of the velocity field, but not on its divergence. As an application, we obtain uniqueness of solutions in the atomic Hardy space, H1, for the scalar conservation law induced by a class of vector fields whose divergence is an unbounded BMO function.

We show that vector fields with exponentially integrable derivatives admit a well defined flow of homeomorphisms X(t,⋅)∈W1,p(t)loc for some p(t)>1, at least for small times. When the field is certain Riesz potential of a bounded function, the result becomes global in time, due to techniques from Geometric Function Theory. The local result also applies to the flows arising from Yudovich solutions to the planar Euler system with bounded vorticity.

We establish the higher fractional differentiability of the solutions to nonlinear elliptic equations in divergence form, i.e., div A(x, Du) = div F, when A is a rho-harmonic type operator, and under the assumption that x bar right arrow A(x, xi) belongs to the critical Besov-Lipschitz space B-n/alpha,q(alpha). We prove that some fractional differentiability assumptions on F transfer to Du with no losses in the natural exponent of integrability. When div F = 0, we show that an analogous extra differentiability property for Du holds true under a Triebel-Lizorkin assumption on the partial map x bar right arrow A(x, xi).

In this paper we show that the homeomorphic solutions to each nonlinear Beltrami equation $\partial_{\bar{z}} f = \mathcal{H}(z, \partial_{z} f)$ generate a two-dimensional manifold of quasiconformal mappings $\mathcal{F}_{\mathcal{H}} \subset W^{1,2}_{\mathrm{loc}}(\mathbb{C})$. Moreover, we show that under regularity assumptions on $\mathcal{H}$, the manifold $\mathcal{F}_{\mathcal{H}}$ defines the structure function $\mathcal{H}$ uniquely.

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 congested transport dynamics arising from a non-autonomous traffic optimization problem. In this setting, we prove one can find an optimal traffic strategy with support on the trajectories of a DiPerna-Lions flow. The proof follows the scheme introduced by Brasco, Carlier and Santambrogio in the autonomous setting, applied to the case of supercritical Sobolev dependence in the spatial variable. This requires both Lipschitz and weighted Sobolev apriori bounds for the minimizers of a class of integral functionals whose ellipticity bounds are satisfied only away from a ball of the gradient variable.

In this paper, we look at quasiconformal solutions phi : C -> C of Beltrami equationspartial derivative z-phi(z) = mu(z)partial derivative(z)phi(z),where mu is an element of L-infinity (C) is compactly supported on If, and parallel to mu parallel to (infinity) < 1 and belongs to the fractional Sobolev space W-alpha,W-2/alpha(C). Our main result states thatlog partial derivative(z)phi is an element of W-alpha,W-2/alpha(C)whenever alpha >= 1/2. Our method relies on an n-dimensional result, which asserts the compactness of the commutator[b, (-Delta)(beta/2) : np/Ln-beta p (R-n) -> L-p(R-n)between the fractional laplacian (-Delta)(beta/2) and any symbol b is an element of W-beta,W-n/beta (R-n), provided that 1 < p < n/beta.

We provide Schauder estimates for nonlinear Beltrami equations and lower bounds of the Jacobians for homeomorphic solutions. The results were announced in [1] but here we give detailed proofs.

We face the well-posedness of linear transport Cauchy problems {[ ∂ u∂ t + b·∇ u + c u = f (0,T)×ℝ^n; u(0,· )=u_0∈ L^∞ ℝ^n ]. under borderline integrability assumptions on the divergence of the velocity field b . For W^1,1_loc vector fields b satisfying |b(x,t)|/1+|x|∈ L^1(0,T; L^1)+L^1(0,T; L^∞ ) and div b∈ L^1(0,T; L^∞ ) + L^1( 0,T; Exp ( L/log L) ) , we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every γ >1 , we construct an example of a bounded autonomous velocity field b with div (b)∈Exp ( L/log ^γ L) for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.

In this paper, we study flows associated to Sobolev vector fields with subexponentially integrable divergence. Our approach is based on the transport equation following DiPerna-Lions [DPL89]. A key ingredient is to use a quantitative estimate of solutions to the Cauchy problem of transport equation to obtain the regularity of density functions.

In this paper, we look at quasiconformal solutions $\phi:\mathbb{C}\to\mathbb{C}$ of Beltrami equations $$ \partial_{\overline{z}} \phi(z)=\mu(z)\,\partial_z \phi (z). $$ where $\mu\in L^\infty(\mathbb{C})$ is compactly supported on $\mathbb{D}$, $\|\mu\|_\infty<1$ and belongs to the fractional Sobolev space $W^{\alpha, \frac2\alpha}(\mathbb{C})$. Our main result states that $$\log\partial_z\phi \in W^{\alpha, \frac2\alpha}(\mathbb{C})$$ whenever $\alpha>\frac12$. Our method relies on an $n$-dimensional result, which asserts the compactness of the commutator $$[b,(-\Delta)^\frac{\beta}{2}]:L^\frac{np}{n-\beta p}(\mathbb{R}^n)\to L^p(\mathbb{R}^n)$$ between the fractional laplacian $(-\Delta)^\frac\beta2$ and any symbol $b\in W^{\beta,\frac{n}\beta}(\mathbb{R}^n)$, provided that $1

We study mappings of finite distortion whose distortion functions belong to the Lebesgue space L (loc) (p) . We establish a local modulus of continuity estimate for the inverse of such a mapping. As an application, we describe the possible compression of Hausdorff measure under such mappings. We also exhibit examples that describe the extent to which our results are sharp.