### A biharmonic elliptic problem with dependence on the gradient and the Laplacian.

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We prove a strong comparison principle for the solution of the Levi equation $L\left(u\right)=\sum {}_{i=1}{}^{n}\left(\right(1+u{}_{t}{}^{2}\left)\right(u{}_{{x}_{i}{x}_{i}}+u{}_{{y}_{i}{y}_{i}})+(u{}_{{x}_{i}}{}^{2}+u{}_{{y}_{i}}{}^{2})u{}_{tt}+2(u{}_{{y}_{i}}-u{}_{{x}_{i}}u{}_{t})u{}_{{x}_{i}t}-2(u{}_{{x}_{i}}+u{}_{{y}_{i}}u{}_{t})u{}_{{y}_{i}t}+k(x,y,t\left)\right(1+\left|Du\right|{}^{2}){}^{3/2}=0$, applying Bony Propagation Principle.

We study the gradient flow of the L2−norm of the second fundamental form for smooth immersions of two-dimensional surfaces into compact Riemannian manifolds. By analogy with the results obtained in [10] and [11] for the Willmore flow, we prove lifespan estimates in terms of the L2−concentration of the second fundamental form of the initial data and we show the existence of blowup limits. Under special condition both on the initial data and on the target manifold, we prove a long time existence result...

In this paper we study the finite element approximations to the Sobolev and viscoelasticity type equations and present a direct analysis for global superconvergence for these problems, without using Ritz projection or its modified forms.

Hörmander’s famous Fourier multiplier theorem ensures the ${L}_{p}$-boundedness of $F(-{\Delta}_{\mathbb{R}}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s\>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the ${L}_{p}$-boundedness of $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta}_{\mathbb{R}}+{x}^{2}$ shows that in general $s\>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper,...

It is shown that the partial differential operator $P\left(D\right)=\partial \u2074/\partial x\u2074-\partial \xb2/\partial y\xb2+i\partial /\partial z:{\Gamma}^{d}\left(\mathbb{R}\xb3\right)\to {\Gamma}^{d}\left(\mathbb{R}\xb3\right)$ is surjective if 1 ≤ d < 2 or d ≥ 6 and not surjective for 2 ≤ d < 6.