Generalized dirichlet problem in nonlinear potential theory

The operator extending the classical solution of the Dirichlet problem for the quasilinear elliptic equation divA(x,▽u)=0 akin to thep-Laplace equation is shown to be unique providedA obeys a specific order principle. The Keldych lemma is also generalized to this nonlinear setting.

Mappings of finite distortion: The zero set of the Jacobian

This paper is part of our program to establish the fundamentals of the theory of mappings of finite distortion [6], [1], [8], [13], [14], [7] which form a natural generalization of the class of mappings of bounded distortion, also called quasiregular mappings. Let us begin with the definition. We assume that Ω ⊂ Rn is a connected open set. We say that a mapping f : Ω → Rn has finite distortion if:

The Wiener test and potential estimates for quasilinear elliptic equations

Mappings of Finite Distortion:¶Discreteness and Openness

We establish a sharp integrability condition on the partial derivatives of a mapping with L p -integrable distortion for some p>n− 1 to guarantee discreteness and openness. We also show that a mapping with exponentially integrable distortion and integrable Jacobian determinant is either constant or both discrete and open. We give an example demonstrating the preciseness of our criterion.

On functions with derivatives in a Lorentz space

We establish a sharp integrability condition on the partial derivatives of a Sobolev mapping to guarantee that sets of measure zero get mapped to sets of measure zero. This condition is sharp also for continuity and differentiability almost everywhere.

Regularity of the inverse of a Sobolev homeomorphism in space

Let Ω ⊂ Rn be open. Given a homeomorphism of finite distortion with |Df| in the Lorentz space Ln−1, 1 (Ω), we show that and f−1 has finite distortion. A class of counterexamples demonstrating sharpness of the results is constructed.

Mappings of finite distortion: Sharp Orlicz-conditions

We establish continuity, openness and discreteness, and the condition $(N)$ for mappings of finite distortion under minimal integrability assumptions on the distortion.

Pointwise Inequalities for Sobolev Functions on Outward Cuspidal Domains

Abstract We show that the 1st-order Sobolev spaces $W^{1,p}(\Omega _\psi ),$$1<p\leq \infty ,$ on cuspidal symmetric domains $\Omega _\psi $ can be characterized via pointwise inequalities. In particular, they coincide with the Hajłasz–Sobolev spaces $M^{1,p}(\Omega _\psi )$.

A note on Sobolev isometric immersions below W2,2 regularity

Abstract This paper aims to investigate the Hessian of second order Sobolev isometric immersions below the natural W 2 , 2 setting. We show that the Hessian of each coordinate function of a W 2 , p , p 2 , isometric immersion satisfies a low rank property in the almost everywhere sense, in particular, its Gaussian curvature vanishes almost everywhere. Meanwhile, we provide an example of a W 2 , p , p 2 , isometric immersion from a bounded domain of R 2 into R 3 that has multiple singularities.

Approximation by mappings with singular Hessian minors

Let $\Omega\subset\mathbb R^n$ be a Lipschitz domain. Given $1\leq p<k\leq n$ and any $u\in W^{2,p}(\Omega)$ belonging to the little H\"older class $c^{1,\alpha}$, we construct a sequence $u_j$ in the same space with $\operatorname{rank}D^2u_j<k$ almost everywhere such that $u_j\to u$ in $C^{1,\alpha}$ and weakly in $W^{2,p}$. This result is in strong contrast with known regularity behavior of functions in $W^{2,p}$, $p\geq k$, satisfying the same rank inequality.