Narrow Results

We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge–Ampère equations, and investigate their regularity. These types of equations are precisely what is needed in order to construct Kähler–Einstein metrics over irreducible singular Kähler spaces with ample or trivial canonical sheaf and singular Kähler–Einstein metrics over varieties of general type.

The paper concerns the model of a flow of non-Newtonian fluid with nonstandard growth conditions of the Cauchy stress tensor. Contrary to standard power-law type rheology, we propose the formulation with the help of the spatially-dependent convex function. This framework includes e.g. rapidly shear thickening and magnetorheological fluids. We provide the existence of weak solutions. The nonstandard growth conditions yield the analytical formulation of the problem in generalized Orlicz spaces. Basing on the energy equality, we exploit the tools of Young measures.

In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrödinger equations -ε²Δu + V(x)u - ε² [Δ(u²)]u = f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger–Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -ε² [Δ(u²)]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter ε for which is motivated by mathematical models in physics. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.

A version of the Lebesgue differentiation theorem is offered, where the $L^p$ norm is replaced with any rearrangement-invariant norm. Necessary and sufficient conditions for a norm of this kind to support the Lebesgue differentiation theorem are established. In particular, Lorentz, Orlicz and other customary norms for which Lebesgue’s theorem holds are characterized.

Let Ω be a bounded domain in R^N with smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered:

In this paper we investigate the measurable transformations that induce composition operators on L^ϕ-1 spaces and study some properties of these operators.

A variant of the monotone operator method is applied to the study of solvability of nonlinear singular integral equations of Hammerstein type in Orlicz spaces.

In this paper we consider the problem: ∂_tu − Δu = f (u), u(0) = u₀ ∈ exp L^p (ℝ^N), where p > 1 and f : ℝ → ℝ having an exponential growth at infinity with f (0) = 0. We prove local well-posedness in $\exp L_0^p\left({ℝ}^N\right)$ for $f\left(u\right)\sim e^{{\left|u\right|}^q},0<q\le p,\left|u\right|\to \infty$. However, if for some λ > 0, $\underset{s\to \infty }{\lim }\inf \left(f\left(s\right)e^{-\lambda s^p}\right)>0$ then non-existence occurs in exp L^p (ℝ^N ). Under smallness condition on the initial data and for exponential nonlinearity f such that |f(u)| ∼ |u|^m as u → 0, $\frac{N\left(m-1\right)}{2}\ge p$, we show that the solution is global. In particular, p – 1 > 0 sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on m.

We summarize recent lectures devoted to the study of the existence and L^∞-regularity results of some Dirichlet problems associated to equations having degenerated coercivity in the principal part.

In this survey we expose the fundamentals of the strongly nonlinear potential theory and relate some of its components. As applications, we establish a relation between this theory and Partial Differential Equations, and show whether the equation Δ_Au + h = 0 possesses a solution or not, for a fixed function h. Here Δ_A is the A-Laplacian which is the p-Laplacian Δ_p, when the Orlicz space L_A reduces to the Lebesgue space L^p.