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We show that vector fields b whose spatial derivative D x b satisfies a Orlicz summability condition have a spatially continuous representative and are well-posed. For the case of sub-exponential summability, their flows satisfy a Lusin (N) condition in a quantitative form, too. Furthermore, we prove that if D x b satisfies a suitable exponential summability condition then the flow associated to b has Sobolev regularity, without assuming boundedness of div x b. We then apply these results to the representation and Sobolev regularity of weak solutions of the Cauchy problem for the transport and continuity equations. [ABSTRACT FROM AUTHOR]

Existence of two solutions to a parametric singular quasi-linear elliptic problem is proved. The equation is driven by the Φ\Phi -Laplacian operator, and the reaction term can be nonmonotone. The main tools employed are the local minimum theorem and the Mountain pass theorem, together with the truncation technique. Global C1,τ{C}^{1,\tau } regularity of solutions is also investigated, chiefly via a priori estimates and perturbation techniques.

We discuss Meyers-Serrin's type results for smooth approximations of functions b = b (t , x) : R × R n → R m , with convergence of an energy of the form ∫ R ∫ R n w (t , x) φ | D b (t , x) | d x d t , where w > 0 is a suitable weight function, and φ : [ 0 , ∞) → [ 0 , ∞) is a convex function with φ (0) = 0 having exponential or subexponential growth. [ABSTRACT FROM AUTHOR]

Let Ω ⊆ R 2 be a domain, let Φ be a Δ 2 Young function and let f ∈ W 1 , Φ ( Ω , R 2 ) be a homeomorphism between Ω and f ( Ω ) . Then there exists a sequence of diffeomorphisms f k converging to f in the Sobolev–Orlicz space W 1 , Φ ( Ω , R 2 ) . Further for an injective continuous map φ ∈ W 1 , Φ ( ∂ ( − 1 , 1 ) 2 , R 2 ) we find a diffeomorphism in W 1 , Φ ( ( − 1 , 1 ) 2 , R 2 ) that equals φ on the boundary. [ABSTRACT FROM AUTHOR]