A natural approach to the asymptotic mean value property for the pp-Laplacian

Let $1\le p\le\infty$. We show that a function $u\in C(\mathbb R^N)$ is a viscosity solution to the normalized $p$-Laplace equation $\Delta_p^n u(x)=0$ if and only if the asymptotic formula u(x)=\mu_p(\ve,u)(x)+o(\ve^2) holds as \ve\to 0 in the viscosity sense. Here, \mu_p(\ve,u)(x) is the $p$-mean value of $u$ on B_\ve(x) characterized as a unique minimizer of \inf_{\la\in\RR}\nr u-\la\nr_{L^p(B_\ve(x))}. This kind of asymptotic mean value property (AMVP) extends to the case $p=1$ previous (AMVP)'s obtained when \mu_p(\ve,u)(x) is replaced by other kinds of mean values. The natural definition of \mu_p(\ve,u)(x) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of $u$. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic $p$-Laplace equation.Comment: 19 pages, submitte

Optimal embeddings of critical Sobolev-Lorentz-Zygmund spaces

We establish the embedding of the critical Sobolev-Lorentz-Zygmund space (Formula presented.)(ℝn ) into the generalized Morrey space M φr (ℝn ) with an optimal Young function φ. As an application, we obtain the almost Lipschitz continuity for func- tions in (Formula presented.)(ℝn ). O\u27Neil\u27s inequality and its reverse play an essential role in the proofs of the main theorems

On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in RN

In this paper, we consider the attainability of a maximizing problem (Formula presented.)where (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.) and (Formula presented.). The existence of a maximizer for D is closely related to the exponent (Formula presented.). In fact, we show that the value (Formula presented.)is a threshold in terms of the attainability of D. © 2015 Springer-Verlag Berlin Heidelberg発行後1年より全文公

Revisiting an idea of Brézis and Nirenberg

AbstractLet n⩾3 and Ω be a C1 bounded domain in Rn with 0∈∂Ω. Suppose ∂Ω is C2 at 0 and the mean curvature of ∂Ω at 0 is negative, we prove the existence of positive solutions for the equation:(0.1){Δu+λun+2n−2+u2∗(s)−1|x|s=0in Ω,u=0on ∂Ω, where λ>0, 0<s<2, 2∗(s)=2(n−s)n−2 and n⩾4. For n=3, the existence result holds for 0<s<1. Under the same assumption of the domain Ω, for p⩽2∗(s)−1, we also prove the existence of a positive solution for the following equation:(0.2){Δu−λup+u2∗(s)−1|x|s=0in Ω,u=0on ∂Ω, whereλ>0and1⩽p<nn−2

Remarks on the Rellich inequality

We study the Rellich inequalities in the framework of equalities. We present equalities which imply the Rellich inequalities by dropping remainders. This provides a simple and direct understanding of the Rellich inequalities as well as the nonexistence of nontrivial extremisers. © 2016 Springer-Verlag Berlin HeidelbergEmbargo Period 12 month