Existence for evolutionary problems with linear growth by stability methods

Bögelein, Verena; Duzaar, Frank; Schätzler, Leah; Scheven, Christoph

doi:10.1016/j.jde.2018.12.012

Cited by 12 publications

(14 citation statements)

References 28 publications

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“…Gianazza and Klaus [14] showed that variational solutions to the Cauchy-Dirichlet problem for the total variation flow are obtained as the limit as p → 1 of variational solutions to the corresponding problem for the parabolic p-Laplace equation. See also Bögelein, Duzaar, Schätzler and Scheven [9]. Our main result in Theorem 5.1 below shows that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under natural assumptions.…”

Section: Introductionmentioning

confidence: 80%

On the definition of solution to the total variation flow

Kinnunen

Scheven

2022

Calc. Var.

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Section: Introductionmentioning

confidence: 80%

On the definition of solution to the total variation flow

Kinnunen

Scheven

2022

Calc. Var.

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“…For a bounded C 1 -domain and time independent Dirichlet boundary values the authors constructed solutions of the total variation flow as limit of solutions of the parabolic p-Laplacian. The overall proof strategy in the present paper is the same as in [14]. The authors use stability methods to prove the existence of variational solutions to Cauchy-Dirichlet problems of the type ∂ t u − div(D ξ f (x, Du)) = 0 in Ω T , u = g on ∂ par Ω,…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Theorem 2.4, we finally prove the desired variational inequality for general comparison maps. Compared to the case of Dirichlet boundary values treated in [14], througout the present paper the construction of admissible comparison maps is somewhat easier, since we do not have to care about boundary values attained by mollifications.…”

Section: Plan Of the Papermentioning

confidence: 99%

Existence for evolutionary Neumann problems with linear growth by stability results

Schätzler¹

2019

Ann. Acad. Sci. Fenn. Math.

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We are concerned with the Neumann type boundary value problem to parabolic systems ∂ t u − div(D ξ f (x, Du)) = −D u g(x, u), where u is vector-valued, f satisfies a linear growth condition and ξ → f (x, ξ) is convex. We prove that variational solutions of such systems can be approximated by variational solutions to ∂ t u − div(D ξ f p (x, Du)) = −D u g(x, u) with p > 1. This can be interpreted both as a stability and existence result for general flows with linear growth.

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“…The notion of variational solutions was introduced by Lichnewsky and Temam in [45] to study parabolic equations by a variational approach, which has advantages when studying equations with unbalanced growth conditions where Lavrentiev phenomenon can prevent the existence of weak solutions. These methods were greatly refined and expaned upon by the group of Bögelein, Duzaar, Marcellini and coauthors in a series of papers such as [7,20,8,9,10,11,12,62]. The group of Stefanelli has also conducted extensive work on variational methods in the existence theory of evolution equations starting from his early work on De Giorgi conjecture [64] and subsequent works [2,59,60].…”

Section: The Problemmentioning

confidence: 99%