A generalised conical density theorem for unrectifiable sets.

Lorent, Andrew

Displaying similar documents to “A generalised conical density theorem for unrectifiable sets.”

Semilinear equations with exponential nonlinearity and measure data

Daniele Bartolucci, Fabiana Leoni, Luigi Orsina, Augusto C. Ponce (2005)

Annales de l'I.H.P. Analyse non linéaire

Similarity:

Sets of determination for solutions of the Helmholtz equation

Jarmila Ranošová (1997)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $α > 0$ , $λ = {(2 α)}^{- 1 / 2}$ , $S^{n - 1}$ be the $(n - 1)$ -dimensional unit sphere, $σ$ be the surface measure on $S^{n - 1}$ and $h (x) = \int_{S^{n - 1}} e^{λ 〈 x, y 〉} d σ (y)$ . We characterize all subsets $M$ of $ℝ^{n}$ such that $inf_{x \in ℝ^{n}} \frac{u (x)}{h (x)} = inf_{x \in M} \frac{u (x)}{h (x)}$ for every positive solution $u$ of the Helmholtz equation on $ℝ^{n}$ . A closely related problem of representing functions of $L_{1} (S^{n - 1})$ as sums of blocks of the form $e^{λ 〈 x_{k}, . 〉} / h (x_{k})$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

Necessary conditions for weak lower semicontinuity on domains with infinite measure

Stefan Krömer (2010)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

We derive sharp necessary conditions for weak sequential lower semicontinuity of integral functionals on Sobolev spaces, with an integrand which only depends on the gradient of a scalar field over a domain in $ℝ^{N}$ . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value $+ \infty$ .