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View article: Minimization of hyperbolic systems of conservation laws
Minimization of hyperbolic systems of conservation laws Open
We consider the Cauchy problem for a generic hyperbolic system of conservation laws and assume it is provided by a standard Riemann semigroup of solutions, for summable initial data with small total variation. Then, we introduce an integra…
View article: Variational solutions to the total variation flow on metric measure spaces
Variational solutions to the total variation flow on metric measure spaces Open
This dissertation studies existence and regularity properties of functions related to the calculus of variations on metric measure spaces that support a weak Poincaré inequality and doubling measure. The work consists of four articles in w…
View article: Existence of parabolic minimizers to the total variation flow on metric measure spaces
Existence of parabolic minimizers to the total variation flow on metric measure spaces Open
We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $$({\mathcal {X}}, d, \mu )$$ satisfying a doubling condition and su…
View article: On BV functions and essentially bounded divergence-measure fields in metric spaces
On BV functions and essentially bounded divergence-measure fields in metric spaces Open
By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space (\mathbb{X},d,\mu) eq…
View article: Variational solutions to the total variation flow on metric measure spaces
Variational solutions to the total variation flow on metric measure spaces Open
We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincaré inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian s…
View article: Existence of parabolic minimizers to the total variation flow on metric measure spaces
Existence of parabolic minimizers to the total variation flow on metric measure spaces Open
We give an existence proof for variational solutions $u$ associated to the total variation flow. Here, the functions being considered are defined on a metric measure space $(\mathcal{X}, d, μ)$ satisfying a doubling condition and supportin…
View article: Time-smoothing for parabolic variational problems in metric measure spaces
Time-smoothing for parabolic variational problems in metric measure spaces Open
In 2013, Masson and Siljander determined a method to prove that the $p$-minimal upper gradient $g_{f_\varepsilon}$ for the time mollification $f_\varepsilon$, $\varepsilon>0$, of a parabolic Newton-Sobolev function $f\in L^p_\mathrm{loc}(0…
View article: On $BV$ functions and essentially bounded divergence-measure fields in\n metric spaces
On $BV$ functions and essentially bounded divergence-measure fields in\n metric spaces Open
By employing the differential structure recently developed by N. Gigli, we\nfirst give a notion of functions of bounded variation ($BV$) in terms of\nsuitable vector fields on a complete and separable metric measure space\n$(\\mathbb{X},d,…