On the Modulus of Continuity of Solutions to the n-Laplace Equation (original) (raw)

On the Notion of Renormalized Solution to Nonlinear Parabolic Equations with General Measure Data

Journal of Elliptic and Parabolic Equations, 2015

Solutions to the n-Laplace equation with a right-hand side f are considered. We exhibit the largest rearrangement-invariant space to which f has to belong for every local weak solution to be continuous. Moreover, we find the optimal modulus of continuity of solutions when f ranges in classes of rearrangement-invariant spaces, including Lorentz, Lorentz-Zygmund and various standard Orlicz spaces.

Maximally singular solutions of Laplace equations

arXiv (Cornell University), 2018

It is known that there exists an explicit function F in L 2 (Ω), where Ω is a given bounded open subset of R N , such that the corresponding weak solution of the Laplace BVP −∆u = F (x), u ∈ H 1 0 (Ω), is maximally singular; that is, the singular set of u (defined in the Introduction) has the Hausdorff dimension equal to (N − 4) +. This constant is optimal, i.e., the largest possible. Here, we show that much more is true: when N ≥ 5, there exists F ∈ L 2 (Ω) such that the corresponding weak solution has the pointwise concentration of singular set of u, in the sense of the Hausdorff dimension, equal to N − 4 at all points of Ω. We also consider the problem of generating weak solutions with the property of contrast; that is, we construct solutions u that are regular (more specifically, of class C 2,α loc for arbitrary α ∈ (0, 1)) in any prescribed open subset Ωr of Ω, while they are maximally singular in its complement Ω \ Ωr. We indicate several open problems.

Optimization problems in classes of rearrangements for ( p , q )-Laplace equations

2018

The non-homogeneous differential operator ∆p + ∆q is called (p,q)-Laplacian. As observed in [17], it stems from a wide range of important applications including biophysics [10], plasma physics [19], reaction-diffusion equations [7], as well as models of elementary particles [2]. In the last decades there has been a great interest in the investigation of these problems mainly concerning existence and multiplicity of solutions, eigenvalues, ground-state solutions [1, 6, 11]. In the present paper we consider the case f(x, u) = g(x)|u|α−1, where 1 ≤ α < q and g(x) is a measurable bounded non-negative function which is positive in a subset with a positive measure. For v ∈ H 0 (Ω) we define

Continuous rearrangement and symmetry of solutions of elliptic problems

Proceedings Mathematical Sciences, 2000

This work presents new results and applications for the continuous Steiner symmetrization. There are proved some functional inequalities, e.g. for Dirichlet-type integrals and convolutions and also continuity properties in Sobolev spaces W 1Y p. Further it is shown that the local minimizers of some variational problems and the nonnegative solutions of some semilinear elliptic problems in symmetric domains satisfy a weak,`local' kind of symmetry.

On Neumann and Poincare problems for Laplace equation

Analysis and Mathematical Physics, 2016

It is proved the existence of nonclassical solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for a special case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the found solutions have the infinite dimension.

Sobolev type inequalities for rearrangement invariant spaces

Positivity, 2010

In the setting of rearrangement invariant spaces, optimal Sobolev inequalities (via the gradient) are well understood. By means of an alternative functional, we obtain new Sobolev inequalities which are finer than (and not necessarily equivalent to) the ones mentioned above.

New formulas for decreasing rearrangements and a class of Orlicz–Lorentz spaces

Revista Matemática Complutense, 2013

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces Mϕ,w and mϕ,w respectively, introduced earlier in [3] for describing the Köthe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases (ϕ is an Orlicz function and w a decreasing weight). We study these Mϕ,w classes in the most general setting, where they may even not be linear, and identify their Köthe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces Mϕ,w which proves to be the Köthe biduals of the Mϕ,w classes. In the case when the class Mϕ,w is a separable quasi-Banach space, Mϕ,w is its Banach envelope.

Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities

Studia Mathematica, 2011

We study imbeddings of the Sobolev space W m, (Ω) := {u : Ω → R with (∂ α u/∂x α) < ∞ when |α| ≤ m}, in which Ω is a bounded Lipschitz domain in R n , is a rearrangement-invariant (r.i.) norm and 1 ≤ m ≤ n − 1. For such a space we have shown there exist r.i. norms, τ and σ , that are optimal with respect to the inclusions W m, (Ω) ⊂ W m,τ (Ω) ⊂ Lσ (Ω). General formulas for τ and σ are obtained using the K-method of interpolation. These lead to explicit expressions when is a Lorentz Gamma norm or an Orlicz norm.