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Hamilton-Jacobi flows and characterization of solutions of Aronsson equations

Petri Juutinen; Eero Saksman — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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 $r \mapsto {max}_{y \in B_{r} (x)} u (y)$ and $r \mapsto {min}_{y \in B_{r} (x)} u (y)$ , respectively.

Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

Eero Saksman; Tomás Soto — 2017

Analysis and Geometry in Metric Spaces

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...

Weak Compactness of Multiplication Operators on Spaces of Bounded Linear Operators.

Eero Saksman; Hans-Olav Tylli — 1992

Mathematica Scandinavica

Superharmonicity of nonlinear ground states.

Peter Lindqvist; Juan Manfredi; Eero Saksman — 2000

Revista Matemática Iberoamericana

The objective of our note is to prove that, at least for a convex domain, the ground state of the p-Laplacian operator Δ_pu = 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...

Isomorphisms of Royden Type Algebras Over $\mathbb{S}^{1}$

Teresa Radice; Eero Saksman; Gabriella Zecca — 2009

Bollettino dell'Unione Matematica Italiana

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 $\|u\|=\|u\|_{L^{\infty}(\mathbb{S}^{1})}+\left(\iint_{\mathbb{D}}|\nabla\tilde% {u}|^{2}\right)^{1/2}$ 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 $\mathbb{S}^{1}$ .

Remarks on the nonexistence of doubling measures.

Saksman, Eero — 1999

Annales Academiae Scientiarum Fennicae. Mathematica

Representing non-weakly compact operators

Manuel González; Eero Saksman; Hans-Olav Tylli — 1995

Studia Mathematica

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...