Structural Stability of Doubly-Nonlinear Flows

Augusto Visintin

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 3, page 363-391
  • ISSN: 0392-4041

Abstract

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To any maximal monotone operator α : V → 𝒫 ⁢ ( V ) ( V being a real Banach space),in [MR1009594] S. Fitzpatrick associated a lower semicontinuous and convex function f : V × V ′ → ℝ ∪ { + ∞ } such that f ⁢ ( v , v ′ ) ≥ ⟨ v ′ , v ⟩   ∀ ( v , v ′ ) , f ⁢ ( v , v ′ ) = ⟨ v ′ , v ⟩ ⇔ v ′ ∈ α ⁢ ( v ) . On this basis, in this work two classes of doubly-nonlinear evolutionary equations are formulated as minimization principles: D t ⁢ α ⁢ ( u ) - div ⁡ γ → ⁢ ( ∇ ⁡ u ) ∋ h , α ⁢ ( D t ⁢ u ) - div ⁡ γ → ⁢ ( ∇ ⁡ u ) ∋ h ; here α and γ → are maximal monotone mappings, and one of them is assumed to be cyclically monotone. For associated initial- and boundary-value problems, existence of a solution is proved, as well as the stability with respect to variations of the data and of the operators D t , ∇ , α and γ → .

How to cite

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Visintin, Augusto. "Structural Stability of Doubly-Nonlinear Flows." Bollettino dell'Unione Matematica Italiana 4.3 (2011): 363-391. <http://eudml.org/doc/290745>.

@article{Visintin2011,
abstract = {To any maximal monotone operator $\alpha \colon V \to \mathcal\{P\}(V)$ ($V$ being a real Banach space),in [MR1009594] S. Fitzpatrick associated a lower semicontinuous and convex function $f \colon V \times V' \to \mathbb\{R\} \cup \\{+\infty\\}$ such that \begin\{equation*\} \tag\{*\} f(v,v') \geq \langle v', v \rangle \quad \forall (v, v'), \qquad f(v,v') = \langle v', v \rangle \iff v' \in \alpha(v).\end\{equation*\} On this basis, in this work two classes of doubly-nonlinear evolutionary equations are formulated as minimization principles: \begin\{equation*\} \tag\{**\} D\_\{t\}\alpha(u) - \operatorname\{div\} \vec\{\gamma\}(\nabla u) \ni h, \qquad \alpha(D\_\{t\}u) - \operatorname\{div\} \vec\{\gamma\}(\nabla u) \ni h; \end\{equation*\} here $\alpha$ and $\vec\{\gamma\}$ are maximal monotone mappings, and one of them is assumed to be cyclically monotone. For associated initial- and boundary-value problems, existence of a solution is proved, as well as the stability with respect to variations of the data and of the operators $D_\{t\}$, $\nabla$, $\alpha$ and $\vec\{\gamma\}$.},
author = {Visintin, Augusto},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {363-391},
publisher = {Unione Matematica Italiana},
title = {Structural Stability of Doubly-Nonlinear Flows},
url = {http://eudml.org/doc/290745},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Visintin, Augusto
TI - Structural Stability of Doubly-Nonlinear Flows
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/10//
PB - Unione Matematica Italiana
VL - 4
IS - 3
SP - 363
EP - 391
AB - To any maximal monotone operator $\alpha \colon V \to \mathcal{P}(V)$ ($V$ being a real Banach space),in [MR1009594] S. Fitzpatrick associated a lower semicontinuous and convex function $f \colon V \times V' \to \mathbb{R} \cup \{+\infty\}$ such that \begin{equation*} \tag{*} f(v,v') \geq \langle v', v \rangle \quad \forall (v, v'), \qquad f(v,v') = \langle v', v \rangle \iff v' \in \alpha(v).\end{equation*} On this basis, in this work two classes of doubly-nonlinear evolutionary equations are formulated as minimization principles: \begin{equation*} \tag{**} D_{t}\alpha(u) - \operatorname{div} \vec{\gamma}(\nabla u) \ni h, \qquad \alpha(D_{t}u) - \operatorname{div} \vec{\gamma}(\nabla u) \ni h; \end{equation*} here $\alpha$ and $\vec{\gamma}$ are maximal monotone mappings, and one of them is assumed to be cyclically monotone. For associated initial- and boundary-value problems, existence of a solution is proved, as well as the stability with respect to variations of the data and of the operators $D_{t}$, $\nabla$, $\alpha$ and $\vec{\gamma}$.
LA - eng
UR - http://eudml.org/doc/290745
ER -

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