In this note, we verify the conjecture of Barron, Evans and Jensen [3] regarding the characterization of viscosity solutions of general Aronsson equations in terms of the properties of associated forward and backwards Hamilton-Jacobi flows. A special case of this result is analogous to the characterization of infinity harmonic functions in terms of convexity and concavity of the functions $r\mapsto {max}_{y\in {B}_{r}\left(x\right)}u\left(y\right)$ and $r\mapsto {min}_{y\in {B}_{r}\left(x\right)}u\left(y\right)$, respectively.

We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function...

The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operator
Δ_{p}u = div (|∇u|^{p-2} ∇u)
is a superharmonic function, provided that 2 ≤ p ≤ ∞. The ground state of Δ_{p} is the positive solution with boundary values zero of the equation
div(|∇u|^{p-2} ∇u) + λ |u|^{p-2} u = 0
in the bounded domain Ω in the n-dimensional...

Let ${\mathbb{S}}^{1}$ and $\mathbb{D}$ be the unit circle and the unit disc in the plane and let us denote by $\mathcal{A}({\mathbb{S}}^{1})$ the algebra of the complex-valued continuous functions on ${\mathbb{S}}^{1}$ which are traces of functions in the Sobolev class ${W}^{1,2}(\mathbb{D})$. On $\mathcal{A}({\mathbb{S}}^{1})$ we define the following norm $$\parallel u\parallel ={\parallel u\parallel}_{{L}^{\mathrm{\infty}}({\mathbb{S}}^{1})}+{\left({\iint}_{\mathbb{D}}{|\nabla \stackrel{~}{u}|}^{2}\right)}^{1/2}$$ where is the harmonic extension of $u$ to $\mathbb{D}$. We prove that every isomorphism of the functional algebra $\mathcal{A}({\mathbb{S}}^{1})$ is a quasitsymmetric change of variables on ${\mathbb{S}}^{1}$.

For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of ${L}^{1}$ and C(0,1), but R(L(E)/W(E)) identifies isometrically with...

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