Search

In this paper we consider a linear elliptic equation in divergence form $$\begin{aligned} \sum _{i,j}D_j(a_{ij}(x)D_i u )=0 \quad \hbox {in } \Omega . \end{aligned}$$ ∑ i , j D j ( a ij ( x ) D i u ) = 0 in Ω . Assuming the coefficients $$a_{ij}$$ a ij in $$W^{1,n}(\Omega )$$ W 1 , n ( Ω ) with a modulus of continuity satisfying a certain Dini-type continuity condition, we prove that any very weak solution $$u\in L^{n'}_\mathrm{loc}(\Omega )$$ u ∈ L loc n ′ ( Ω ) of (0.1) is actually a weak solution in $$W^{1,2}_\mathrm{loc}(\Omega )$$ W loc 1 , 2 ( Ω ) .

Given a Banach space E with a supremum-type norm induced by a collection of operators, we prove that E is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space B introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual B ⁎ , the biduality result that B 0 ⁎ = B ⁎ and B ⁎ ⁎ = B , and a formula for the distance from an element f ∈ B to B 0 .

We consider pairs of non reflexive Banach spaces $$(E_0, E)$$ such that $$E_0$$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $$(E_0, E)$$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators $$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$ where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $$(B_0, B)$$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $$(L^{\psi }_0, L^{\psi })$$ where $$L^{\psi }_0$$ is the closure of $$L^\infty $$ in the Orlicz space $$L^{\psi }$$ , Marcinkiewicz pairs $$(L^{q, \infty }_0, L^{q, \infty })$$ where $$L^{q, \infty }_0$$ is the closure of $$L^\infty $$ in the Marcinkiewicz weak– $$L^q$$ denoted by $$L^{q, \infty }$$ . More generally, Banach function spaces are considered. The main results are duality formulas of the type 1 $$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$ 2 $$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$ and distance formulas.

In this paper we introduce the class of the inner p-quasiconformal mappings, that are homeomorphisms f : D ⟶ o n t o D , f ∈ W loc 1 , 1 ( D ; D ) , where D ⊂ R 2 is the unit disk, such that there exists a constant K p ≥ 0 for which the following distortion inequality | D f ( x ) | p ≤ K p | J f ( x ) | p − 1 a.e. x ∈ D is satisfied. The study of such mappings is motivated by the fact that their inverses satisfy the distortion inequality introduced in [11] . Here we give a characterization of them so that their components solve a suitable uniformly elliptic p-harmonic system. Moreover, for mappings satisfying the previous distortion inequality with K p = K p , f ( x ) not necessarily constant, we identify the homeomorphism f whose p-distortion function K p , f ( x ) is minimal in L 1 norm.

This note is concerned with the Poisson equation in the unit disk of the complex plane. Our setting is in the Sobolev space W1;Þ with exponent 1 < p < infty. Such a setting with p eq2 is referred to as beyond the natural domain of definition. The novelty lies in the use of a singular integral called Beurling Transform.

Let Ω ⊂ R n be an open set. We show that for a continuous mapping f ∈ W 1 , n − 1 ( Ω , R n ) with J f ∈ L 1 ( Ω ) the validity of the Lusin (N) condition implies that the distributional Jacobian equals to the pointwise Jacobian.

Let $$N\ge 3$$ . We construct a homeomorphism $$f$$ in the Sobolev space $$W^{1,1}((0,1)^N,(0,1)^N)$$ such that $$f^{-1}\in W^{1,1}((0,1)^N,(0,1)^N)$$ , $$J_f=0$$ a.e. and $$J_{f^{-1}}=0$$ a.e. It follows that $$f$$ maps a set of full measure to a null set and a remaining null set to a set of full measure.We also show that such a pathological homeomorphism cannot exist in dimension $$N=2$$ or with higher regularity $$f\in W^{1,N-1}$$ .

A chain rule for an a.e. approximately differentiable homeomorphism $${{f: \Omega \subset \mathbb{R}^{n} \xrightarrow{\rm onto}} \Omega^{\prime} \subset \mathbb{R}^{n}}$$ is proved. Namely, there exists a Borel set $${{B \subset \mathcal{R}_{f}}}$$ , where $$\mathcal{R}_{f} = \{x \in \Omega : f\, {\rm is\,approximately\,differentiable\,at}\,x\,{\rm and}\,J_{f}(x) \neq 0\},$$ such that $${{\mid \mathcal{R}_{f} \setminus B \mid = 0}}$$ , $${{f(B) \subset \mathcal{R}_{f-1}}}$$ and J f-1(f(x))J f (x) = 1 for all $${{x \in B}}$$ . In general, it is not true that $${{f(\mathcal{R}_{f}) \subset \mathcal{R}_{f-1}}}$$ . Indeed, there exists a homeomorphism $${f_{0}: \Omega \subset \mathbb{R}^{n} \xrightarrow {\rm onto} \Omega^{\prime} \subset \mathbb{R}^{n}}$$ such that f 0 is approximately differentiable at x 0 with $${{J_{f0}(x_{0}) \neq 0}}$$ and $${{{f_{0}^{-1}}}}$$ is not approximately differentiable at f 0(x 0). We give various conditions to guarantee that a homeomorphism preserves density points.

Function spaces that are slightly larger than the Lebesgue Lp(Ω) spaces (even larger than the Marcinkiewicz Lp,∞(Ω) spaces) have been introduced by Iwaniec and Sbordone [Arch. Ration. Mech. Anal. 119 (1992), 129–143] in connection with integrability properties of the Jacobian. These are the grand Lebesgue spaces Lp)(Ω). In this survey we collect a number of results which prove that these spaces are useful in various classical settings of geometric function theory and partial differential equations (PDEs).

We provide regularity properties for the inverse map f^-1 under suitable assumptions on q-distorsion function of f, in bounded domains of R^2.

We give sharp conditions under which the composition of two homeomorphisms of finite distortion is of finite distortion and has integrable distortion. As an application, we obtain a generalization of the classical uniqueness theorem of homeomorphic solution to the measurable Riemann mapping problem.