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Register a new userAsymptotic Estimates for Perturbed Scaiar Curvature Equation
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Samy Skander Bahoura
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We have an idea on the influence of a nonlinear term (tending to 0) on the prescribed scalar curvature equation to have an uniform estimate.
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In this paper, we derive a priori interior Hessian estimates for Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives.
We introduce a new model of the logarithmic type of wave-like equation with a nonlocal logarithmic damping mechanism, which is rather weakly effective as compared with frequently studied fractional damping cases. We consider the Cauchy problem for this new model in the whole space, and study the asymptotic profile and optimal decay and/or blowup rates of solutions as time goes to infinity in L^{2}-sense. The operator L considered in this paper was used to dissipate the solutions of the wave equation in the paper studied by Charao-Ikehata in 2020, and in the low frequency parameters the principal part of the equation and the damping term is rather weakly effective than those of well-studied power type operators.
We derive local estimates of positive solutions to the conformal $Q$-curvature equation $$ (-Delta)^m u = K(x) u^{frac{n+2m}{n-2m}} ~~~~~~ in ~ Omega backslash Lambda $$ near their singular set $Lambda$, where $Omega subset mathbb{R}^n$ is an open set, $K(x)$ is a positive continuous function on $Omega$, $Lambda$ is a closed subset of $mathbb{R}^n$, $2 leq m < n/2$ and $m$ is an integer. Under certain flatness conditions at critical points of $K$ on $Lambda$, we prove that $u(x) leq C [{dist}(x, Lambda)]^{-(n-2m)/2}$ when the upper Minkowski dimension of $Lambda$ is less than $(n-2m)/2$.
Alexandrovs soap bubble theorem asserts that spheres are the only connected closed embedded hypersurfaces in the Euclidean space with constant mean curvature. The theorem can be extended to space forms and it holds for more general functions of the principal curvatures. In this short review, we discuss quantitative stability results regarding Alexandrovs theorem which have been obtained by the author in recent years. In particular, we consider hypersurfaces having mean curvature close to a constant and we quantitatively describe the proximity to a single sphere or to a collection of tangent spheres in terms of the oscillation of the mean curvature. Moreover, we also consider the problem in a non local setting, and we show that the non local effect gives a stronger rigidity to the problem and prevents the appearance of bubbling.
We study in this series of articles the Kardar-Parisi-Zhang (KPZ) equation $$ partial_t h(t,x)= uDelta h(t,x)+lambda V(| abla h(t,x)|) +sqrt{D}, eta(t,x), qquad xin{mathbb{R}}^d $$ in $dge 1$ dimensions. The forcing term $eta$ in the right-hand side is a regularized white noise. The deposition rate $V$ is assumed to be isotropic and convex. Assuming $V(0)ge 0$, one finds $V(| abla h|)ltimes | abla h|^2$ for small gradients, yielding the equation which is most commonly used in the literature. The present article is dedicated to existence results and PDE estimates for the solution. Our results extend in a non-trivial way those previously obtained for the noiseless equation. We prove in particular a comparison principle for sub- and supersolutions of the KPZ equation in new functional spaces containing unbounded functions, implying existence and uniqueness. These new functional spaces made up of functions with locally bounded averages, generically called ${cal W}$-spaces thereafter, and which may be of interest for the study of parabolic equations in general, allow local or pointwise estimates. The comparison to the linear heat equation through a Cole-Hopf transform is an essential ingredient in the proofs, and our results are accordingly valid only for a function $V$ with at most quadratic growth at infinity.
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