Uniform Estimates in Periodic Homogenization of Fully Nonlinear Elliptic Equations

This article is concerned with uniform $$C^{1,\alpha }$$ C 1 , α and $$C^{1,1}$$ C 1 , 1 estimates in periodic homogenization of fully nonlinear elliptic equations. The analysis is based on the compactness method, which involves linearization of the operator at each approximation step. Due to the nonlinearity of the equations, the linearized operators involve the Hessian of correctors, which appear in the previous step. The involvement of the Hessian of the correctors deteriorates the regularity of the linearized operator, and sometimes even changes its oscillating pattern. These issues are resolved with new approximation techniques, which yield a precise decomposition of the regular part and the irregular part of the homogenization process, along with a uniform control of the Hessian of the correctors in an intermediate level. The approximation techniques are even new in the context of linear equations. Our argument can be applied not only to concave operators, but also to certain class of non-concave operators.

Neumann problem for Monge-Ampere type equations revisited.

This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

Weakly convex hypersurfaces of pseudo-Euclidean spaces satisfying the condition LkHk+1 = λHk+1

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

Construction of excited multi-solitons for the 5D energy-critical wave equation

For the 5D energy-critical wave equation, we construct excited [Formula: see text]-solitons with collinear speeds, i.e. solutions [Formula: see text] of the equation such that [Formula: see text] where for [Formula: see text], [Formula: see text] is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel–Merle [Construction of multi-solitons for the energy-critical wave equation in dimension 5, Arch. Ration. Mech. Anal. 222(3) (2016) 1113–1160] and Côte–Martel [Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] developed for the energy-critical wave and nonlinear Klein–Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.

Local stabilization of viscous Burgers equation with memory

<p style='text-indent:20px;'>In this article, we study the local stabilization of the viscous Burgers equation with memory around the steady state zero using localized interior controls. We first consider the linearized equation around zero which corresponds to a system coupled between a parabolic equation and an ODE. We show the feedback stabilization of the system with any exponential decay <inline-formula><tex-math id="M1">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \omega\in (0, \omega_0) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M3">\begin{document}$ \omega_0&gt;0 $\end{document}</tex-math></inline-formula>, using a finite dimensional localized interior control. The control is obtained from the solution of a suitable degenerate Riccati equation. We do an explicit analysis of the spectrum of the corresponding linearized operator. In fact, <inline-formula><tex-math id="M4">\begin{document}$ \omega_0 $\end{document}</tex-math></inline-formula> is the unique accumulation point of the spectrum of the operator. We also show that the system is not stabilizable with exponential decay <inline-formula><tex-math id="M5">\begin{document}$ -\omega $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \omega&gt;\omega_0 $\end{document}</tex-math></inline-formula>, using any <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-control. Finally, we obtain the local stabilization result for the nonlinear system by means of the feedback control stabilizing the linearized system using the Banach fixed point theorem.</p>

On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

A new formulation using the Schur complement for the numerical existence proof of solutions to elliptic problems: without direct estimation for an inverse of the linearized operator

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite-dimensional Newton-type fixed point equation $$w = - {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) + {\mathcal {L}}^{-1} {\mathcal {G}}(w)$$ w = - L - 1 F ( u ^ ) + L - 1 G ( w ) , where $${\mathcal {L}}$$ L is a linearized operator, $${\mathcal {F}}(\hat{u})$$ F ( u ^ ) is a residual, and $${\mathcal {G}}(w)$$ G ( w ) is a nonlinear term. Therefore, the estimations of $$\Vert {\mathcal {L}}^{-1} {\mathcal {F}}(\hat{u}) \Vert $$ ‖ L - 1 F ( u ^ ) ‖ and $$\Vert {\mathcal {L}}^{-1}{\mathcal {G}}(w) \Vert $$ ‖ L - 1 G ( w ) ‖ play major roles in the verification procedures . In this paper, using a similar concept to block Gaussian elimination and its corresponding ‘Schur complement’ for matrix problems, we represent the inverse operator $${\mathcal {L}}^{-1}$$ L - 1 as an infinite-dimensional operator matrix that can be decomposed into two parts: finite-dimensional and infinite-dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, which enables a more efficient verification procedure compared with existing Nakao’s methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as $${\mathcal {L}}^{-1}$$ L - 1 are presented in the “Appendix”.

Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions.

Spectrum of the M5-traveling waves

In this paper, we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M5-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in L2(ℝ; ℂ3) as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space Lw2(ℝ; ℂ3) with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space Lwα2(ℝ; ℂ2) × Lwε2(ℝ; ℂ) the essential spectrum lies on {Reλ<0}, outside the imaginary axis.