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

Sets of determination for solutions of the Helmholtz equation

Jarmila Ranošová (1997)

Commentationes Mathematicae Universitatis Carolinae

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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 ) = S n - 1 e λ x , y d σ ( y ) . We characterize all subsets M of n such that inf x n u ( x ) h ( x ) = inf x M u ( x ) h ( x ) for every positive solution u of the Helmholtz equation on 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

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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 N . An emphasis is put on domains with infinite measure, and the integrand is allowed to assume the value + .