# Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to the Distance Function to the Boundary

Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to... In this article, we are interested in the existence, uniqueness and regularity of the solution of the linear elasticity system. More precisely, the quasi-static elasticity system. In the first part, we study the existence of a weak solution and the regularity in the space $$W^{1, p}_0(\Omega ),\ \forall p \in ]1, +\infty [$$ W 0 1 , p ( Ω ) , ∀ p ∈ ] 1 , + ∞ [ for a p-integrable source function. In the second part, the very weak solution is introduced which can be considered when the second member is a function with a very weak solution, for example, a locally integrable function. Such source functions lead to a lack of regularity for the solution in the fact that existence in classical spaces is no longer assured. So, to overcome this difficulty, the strategy consists in approaching it by another more regular problem “converging” towards the initial problem “in a direction to be specified”. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

# Study of the Regularity of Solutions for a Elasticity System with Integrable Data with Respect to the Distance Function to the Boundary

, Volume 15 (2) – Feb 22, 2018
17 pages

/lp/springer_journal/study-of-the-regularity-of-solutions-for-a-elasticity-system-with-carsWO0hxK
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-018-1088-x
Publisher site
See Article on Publisher Site

### Abstract

In this article, we are interested in the existence, uniqueness and regularity of the solution of the linear elasticity system. More precisely, the quasi-static elasticity system. In the first part, we study the existence of a weak solution and the regularity in the space $$W^{1, p}_0(\Omega ),\ \forall p \in ]1, +\infty [$$ W 0 1 , p ( Ω ) , ∀ p ∈ ] 1 , + ∞ [ for a p-integrable source function. In the second part, the very weak solution is introduced which can be considered when the second member is a function with a very weak solution, for example, a locally integrable function. Such source functions lead to a lack of regularity for the solution in the fact that existence in classical spaces is no longer assured. So, to overcome this difficulty, the strategy consists in approaching it by another more regular problem “converging” towards the initial problem “in a direction to be specified”.

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Feb 22, 2018

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