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Boundary Estimates for Solutions to Parabolic Equations [Elektronisk resurs]

Sande, Olow 1983- (författare)
Nyström, Kaj (preses)
Klimek, Maciej (preses)
Kinnunen, Juha (opponent)
Uppsala universitet Teknisk-naturvetenskapliga vetenskapsområdet (utgivare)
Uppsala Department of Mathematics 2016
Engelska 50
Serie: Uppsala Dissertations in Mathematics 1401-2049
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  • E-bokAvhandling(Diss. (sammanfattning) Uppsala : Uppsala universitet, 2016)
Sammanfattning Ämnesord
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  • This thesis concerns the boundary behavior of solutions to parabolic equations. It consists of a comprehensive summary and four scientific papers. The equations concerned are different generalizations of the heat equation. Paper I concerns the solutions to non-linear parabolic equations with linear growth. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the Riesz measure associated with such solutions, and the Hölder continuityof the quotient of two such solutions up to the boundary. Paper 2 concerns the solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a Muckenhoupt weight of class 1+2/n. For non-negative solutions that vanish continuously on the lateral boundary of an NTA cylinder the following main results are established: a backward Harnack inequality, the doubling property for the parabolic measure, and the Hölder continuity of the quotient of two such solutions up to the boundary. Paper 3 concerns a fractional heat equation. The first main result is that a solution to the fractional heat equation in Euclidean space of dimension n can be extended as a solution to a certain linear degenerate parabolic equation in the upper half space of dimension n+1. The second main result is the Hölder continuity of quotients of two non-negative solutions that vanish continuously on the latteral boundary of a Lipschitz domain. Paper 4 concerns the solutions to uniformly parabolic linear equations with complex coefficients. The first main result is that under certain assumptions on the opperator the bounds for the single layer potentials associated to the opperator are bounded. The second main result is that these bounds always hold if the opperator is realvalued and symmetric. 

Ämnesord

Natural Sciences  (hsv)
Mathematics  (hsv)
Naturvetenskap  (hsv)
Matematik  (hsv)
Natural Sciences  (hsv)
Mathematics  (hsv)
Mathematical Analysis  (hsv)
Naturvetenskap  (hsv)
Matematik  (hsv)
Matematisk analys  (hsv)
Mathematics  (uu)
Matematik  (uu)

Indexterm och SAB-rubrik

Uniformly parabolic equations
non-linear parabolic equations
linear growth
degenerate parabolic equations
fractional heat equations
complex coefficients
Lipschitz domain
NTA domain
boundary behaviour
boundary Harnack
parabolic measure
Riesz measure
Dirichlet to Neumann map
single layer potentials.
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