A notion of total variation depending on a metric with discontinuous coefficients

Abstract

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Lower semicontinuity and relaxation results in BV for integral functionals with BV integrands

“…Indeed, examples can be given of functionals of the type (1.1) whose relaxed functional is given by a functional of the type F where the integrand f is not the product of two functions of x and ξ (see [1], Sect. 8).…”

Local and nonlocal weighted $p$-Laplacian evolution equations with Neumann boundary conditions

“…To define the differential operator div g Du |Du| we need to recall the concept of total variation with respect to an anisotropy (see [1], [14] and [24]). We say that a function φ : Ω × R N → [0, ∞) is a metric integrand if φ is a Borel function satisfying the conditions for a.e.…”

“…Moreover, if φ is coercive in Ω, that is, there exist λ > 0 such that λ ξ ≤ φ(x, ξ) for all x ∈ Ω and for every ξ ∈ R N , and continuous in second variable, in [1] it is proved that BV φ (Ω) = BV (Ω) and…”

An optimal anisotropic Poincaré inequality for convex domains

“…Keywords: anisotropic Laplacian, first eigenvalue, gradient estimate, optimal Poincaré inequality. 305 The anisotropic Laplacian has been studied by many mathematicians, in the context of both Finsler geometry (see, for example, [Amar and Bellettini 1994;Ge and Shen 2001;Ohta 2009;Ohta and Sturm 2011;Shen 2001]) and quasilinear PDE (see, for example, [Alvino et al 1997;Belloni et al 2003;Ferone and Kawohl 2009;Wang and Xia 2011b;2011a;2012]). Particularly, many problems related to the first eigenvalue of the anisotropic Laplacian have already been considered in [Belloni et al 2003;Ge and Shen 2001;Kawohl 2011;Ohta 2009;Wang and Xia 2011a].…”