On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle

2021 ◽  
pp. 1-21
Author(s):  
Paolo Piersanti
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Obstacle Problem ◽  
Shallow Shell ◽  
Shallow Shells ◽  
Interior Regularity ◽  
Order Elliptic Problem ◽  
Higher Differentiability ◽  
Differentiability Properties

In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.

scholarly journals On the improved interior regularity of a boundary value problem modelling the displacement of a linearly elastic elliptic membrane shell subject to an obstacle

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Paolo Piersanti
Keyword(s):  
Boundary Value Problem ◽  
Obstacle Problem ◽  
Boundary Value ◽  
Membrane Shell ◽  
Interior Regularity ◽  
Membrane Shells ◽  
Higher Differentiability ◽  
Differentiability Properties

<p style='text-indent:20px;'>In this paper we show that the solution of an obstacle problem for linearly elastic elliptic membrane shells enjoys higher differentiability properties in the interior of the domain where it is defined.</p>

scholarly journals Periodic and asymptotically periodic fourth-order Schrödinger equations with critical and subcritical growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Edcarlos D. Silva ◽  
Marcos L. M. Carvalho ◽  
Claudiney Goulart
Keyword(s):  
Fourth Order ◽  
Elliptic Problem ◽  
Principal Part ◽  
Existence Of Solutions ◽  
Energy Functional ◽  
Critical Nonlinearities ◽  
Asymptotically Periodic ◽  
Whole Space ◽  
Continuous Potential ◽  
Order Elliptic Problem

<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha &gt; 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

scholarly journals Properties of the extremal solution for a fourth-order elliptic problem

2012 ◽  
Vol 142 (5) ◽  
pp. 1051-1069 ◽  
Author(s):  
Baishun Lai ◽  
Zhuoran Du
Keyword(s):  
Weak Solution ◽  
Fourth Order ◽  
Elliptic Problem ◽  
Open Problem ◽  
Extremal Solution ◽  
Normal Vector ◽  
Unique Weak Solution ◽  
Unit Normal Vector ◽  
Order Elliptic Problem ◽  
Image Position

Let λ* > 0 denote the largest possible value of λ such that the systemhas a solution, where $\mathbb{B}$ is the unit ball in ℝn centred at the origin, p > 1 and n is the exterior unit normal vector. We show that for λ = λ* this problem possesses a unique weak solution u*, called the extremal solution. We prove that u* is singular when n ≥ 13 for p large enough and actually solve part of the open problem which Dávila et al. left unsolved.

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