“…We may repeat the argument of [10], [12] to claim that u ǫ converges to u in L p (0, T ; L q (Ω)), p, q ∈ (1, ∞), (4.8)…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
confidence: 99%
“…We know from [10] that u ǫ,γ x converges to u ǫ x strongly in L p (0, T ; L q (T)), p ≥ 1 and a.e. in Q T , (4.5) thus G(u ǫ x (·, t)) ≤ G(u ǫ 0,x ).…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in W 1,1 . In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.
“…We may repeat the argument of [10], [12] to claim that u ǫ converges to u in L p (0, T ; L q (Ω)), p, q ∈ (1, ∞), (4.8)…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
confidence: 99%
“…We know from [10] that u ǫ,γ x converges to u ǫ x strongly in L p (0, T ; L q (T)), p ≥ 1 and a.e. in Q T , (4.5) thus G(u ǫ x (·, t)) ≤ G(u ǫ 0,x ).…”
Section: Integrability Of the Derivative Of Solutions To The Evolutiomentioning
We study integrability of the derivative of solutions to a singular one-dimensional parabolic equation with initial data in W 1,1 . In order to avoid additional difficulties we consider only the periodic boundary conditions. The problem we study is a gradient flow of a convex, linear growth variational functional. We also prove a similar result for the elliptic companion problem, i.e. the time semidiscretization.
“…Here, by a weak solution u to (1.5) we mean such a function that u x ∈ L ∞ (0, T ; BV (I )), (1.7) and the following identity holds We do not present the proof of the existence here, it will appear in [20]. We apply there the same technique as used in [19,Sect.…”
mentioning
confidence: 90%
“…We stress that we relegate the technical issues of the existence of a solution to another paper, see [20]. A stationary version of (1.5) has been considered in [17].…”
We study qualitative properties of solutions to a monodimensional problemwith the Dirichlet boundary conditions. Such a system presents a key analytical challenge coming from the examination of models of anisotropic phenomena like crystal growth. Our analysis concentrates on the properties of facets-flat regions of solutions-typical for this type of problems.
“…In case of (1.3), we have W (p) = |p|, this is why it is called the total variation flow. Equation (1.3) has been studied quite extensively, [15], [19], [24], [25], [30], [32], [35]. These authors did not allow jumps in the initial datum u 0 .…”
We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in BV , which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga-Giga.
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