A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

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

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.

On a bi-nonlocal fourth order elliptic problem

This paper is aiming at obtaining weak solution for a bi-nonlocal fourth order elliptic problem with Navier boundary condition. Our approach is based on variational methods and critical point theory.

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

<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>

A Paneitz–Branson type equation with Neumann boundary conditions

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely, we prove that the best constant is achieved by a nonconstant solution of the associated fourth order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.