Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity

Knops, R. J.; Stuart, Charles Alexander

doi:10.1007/bf00281557

Cited by 132 publications

(71 citation statements)

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“…Several other results also cover the case Ω = R d , see [67]. There are also numerous papers in the case of more general non-linearities, including, for instance, functions of x, u, and ∇u (see [42]), or more general bifurcation problems than the ones considered in this paper. It is out of the scope of this introduction to review all of them.…”

Section: Introductionmentioning

confidence: 79%

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Dolbeault

Stańczy

2009

Ann. Henri Poincaré

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Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohožaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.

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Section: Introductionmentioning

confidence: 79%

Non-Existence and Uniqueness Results for Supercritical Semilinear Elliptic Equations

Dolbeault

Stańczy

2009

Ann. Henri Poincaré

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“…However, the last term on the right can be extremely singular as r → 0, and I do not know any interesting examples for which I can usefully estimate this expression. Knops & Stuart [11] have used (1.7) to prove the uniqueness of smooth solutions to the Euler-Lagrange equation for quasi-convex F, subject to linear boundary conditions. The forthcoming paper by Kristensen & Mingione [12] presents some further interesting applications of monotonicity formulae to quasi-convex integrands.…”

Section: (C) Concluding Commentsmentioning

confidence: 99%

Monotonicity formulae for variational problems

Evans

2013

Phil. Trans. R. Soc. A.

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“…However, verifying the existence and the uniqueness of the radially symmetric solution for a given elastic material might be rather difficult, so the main result of this article can not be directly used as a condition disallowing cavitation, like the ones presented in [4], [3] or [7]. Section 2 is concerned with the statement of the radially symmetric boundaryvalue problem, in both its weak and strong forms.…”

Section: Introductionmentioning

confidence: 99%