Size of the zero set of solutions of elliptic PDEs near the boundary of Lipschitz domains with small Lipschitz constant
Gallegos, Josep M. (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data:	2023
Resum:	Let Ω ⊂ R be a C domain or, more generally, a Lipschitz domain with small Lipschitz constant and A(x) be a d× d uniformly elliptic, symmetric matrix with Lipschitz coefficients. Assume u is harmonic in Ω, or with greater generality u solves div (A(x) ∇ u) = 0 in Ω, and u vanishes on Σ = ∂Ω ∩ B for some ball B. We study the dimension of the singular set of u in Σ, in particular we show that there is a countable family of open balls (Bi)i such that u|Bi∩Ω does not change sign and K\ ⋃ B has Minkowski dimension smaller than d- 1 - ϵ for any compact K⊂ Σ. We also find upper bounds for the (d- 1) -dimensional Hausdorff measure of the zero set of u in balls intersecting Σ in terms of the frequency. As a consequence, we prove a new unique continuation principle at the boundary for this class of functions and show that the order of vanishing at all points of Σ is bounded except for a set of Hausdorff dimension at most d- 1 - ϵ.
Ajuts:	European Commission 101018680 Agencia Estatal de Investigación PID2020-114167GB-I00
Nota:	Altres ajuts: acords transformatius de la UAB
Drets:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Publicat a:	Calculus of Variations and Partial Differential Equations, Vol. 62, Issue 4 (May 2023) , art. 113, ISSN 1432-0835

DOI: 10.1007/s00526-022-02426-x

52 p, 769.5 KB

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