Let $u(t,x)$ be a solution of the heat equation in $\mathbb{R}^n$. Then, each $k-$th derivative also solves the heat equation and satisfies a maximum principle, the largest $k-$th derivative of $u(t,x)$ cannot be larger than the largest $k-$th derivative of $u(0,x)$. We prove an analogous statement for the solution of the heat equation on bounded domains $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions. As an application, we give a new and fairly elementary proof of the sharp growth of the second derivatives of Laplacian eigenfunction $-Δϕ_k = λ_k ϕ_k$ with Dirichlet conditions on smooth domains $Ω\subset \mathbb{R}^n$.
In this paper, we are inspired by Ngô, Nguyen and Phan's [15] study of the pointwise inequality for positive $C^{4}$-solutions of biharmonic equations with negative exponent by using the growth condition of solutions. They propose an open question of whether the growth condition is necessary to obtain the pointwise inequality. We give a positive answer to this open question. We establish the following local pointwise inequality $$-\frac{Δu}{u}+α\frac{|\nabla u|^{2}}{u^{2}}+βu^{-\frac{q+1}{2}}\leq\frac{C}{R^{2}}$$ for positive $C^{4}$-solutions of the biharmonic equations with negative exponent…
A sharp pointwise differential inequality for vectorial second-order partial differential operators, with Uhlenbeck structure, is offered. As a consequence, optimal second-order regularity properties of solutions to nonlinear elliptic systems in domains in $\mathbb{R}^n$ are derived. Both local and global estimates are established. Minimal assumptions on the boundary of the domain are required for the latter. In the special case of the $p$-Laplace system, our conclusions broaden the range of the admissible values of the exponent $p$ previously known.
We obtain a priori local pointwise second derivative estimates for solutions $u$ to a class of augmented Hessian equations on Riemannian manifolds, in terms of the $C^1$ norm and certain $W^{2,p}$ norms of $u$. We consider the case that no structural assumptions are imposed on either the augmenting term or the right hand side of the equation, and the case where these terms are convex in the gradient variable. In the latter case, under an additional ellipticity condition we prove that the dependence on any $W^{2,p}$ norm can be dropped. Our results are derived using integral estimates.
We prove local pointwise second derivative estimates for positive $W^{2,p}$ solutions to the $σ_k$-Yamabe equation on Euclidean domains, addressing both the positive and negative cases. Generalisations for augmented Hessian equations are also considered.
In this study, we consider a linearized compressible flow structure interaction PDE model for which the interaction interface is under the effect of material derivative term. While the linearization takes place around a constant pressure and density components in structure equation, the flow linearization is taken with respect to a non-zero, fixed, variable ambient vector field. This process produces extra "convective derivative" and "material derivative" terms which causes the coupled system to be nondissipative. We analyze the long time dynamics in the sense of asymptotic (strong) stability …
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type problems and one phase problems.
We prove a sharp Beckner's inequality for axially symmetric functions on $S^4$. The key ingredients of our proof is some pointwise quantitative properties of Gegenbauer polynomials.
The aim of this paper is to establish new pointwise regularity results for solutions to degenerate second order partial differential equations with a Kolmogorov-type operator of the form $$\mathscr{L} :=\sum_{i,j=1}^m \partial^2_{x_i x_j } +\sum_{i,j=1}^N b_{ij}x_j\partial_{x_i}-\partial_t, $$ where $(x,t) \in \mathbb{R}^{N+1}$, $1 \leq m \le N$ and the matrix $B:=(b_{ij})_{i,j=1,\ldots,N}$ has real constant entries. In particular, we show that if the modulus of $L^p$-mean oscillation of $\mathscr{L} u$ at the origin is Dini, then the origin is a Lebesgue point of continuity in $L^p$ average f…
In this paper, we apply blow-up analysis and Liouville type theorems to study pointwise a priori estimates for some quasilinear equations with p-Laplace operator. We first obtain pointwise interior estimates for the gradient of p-harmonic function, i.e., the solution of $Δ_{p}u=0,\ x\inΩ$, which extends the well-established results of the interior estimates of the gradient of harmonic function. We then get singularity and decay estimates of the sign changing solution of Lane-Emden-Fowler type p-Laplace equation $-Δ_{p}u=|u|^{λ-1}u, \ x\inΩ$, which are then generalized for the equation with gen…
Uniqueness from pointwise observations in a multi-parameter inverse problem
本文证明了一维反应扩散方程确定多个非常数系数反问题的一个唯一性结果。这种反应扩散方程包括Kolmogorov、Petrovsky和Piskunov的经典模型以及来自生物学的更复杂的模型。当反应项中含有一个非常系数为$μ_k(x),$的$N,$次的未知多项式部分时，我们的结果给出了该多项式部分的唯一性的一个充分条件.这个充分条件只涉及逐点测量的反应扩散方程的解决方案$u$和它的空间导数$\partial u / \partial x$在一个单一的点$x_0,$在一个时间间隔$(0,ε).$。除了这个唯一性的结果，我们给了几个反例的唯一性，强调我们的假设的最优性。最后，在特定的情况下，N=2和$N=3,$，我们表明，这种逐点测量可以允许一个有效的数值确定未知的多项式反应项。
We consider Lane-Emden ground states with polytropic index $0\leq q-1\leq 1$, that is, minimizers of the Dirichlet integral among $L^q$-normalized functions. Our main result is a sharp lower bound on the $L^2$-norm of the normal derivative in terms of the energy, which implies a corresponding isoperimetric inequality. Our bound holds for arbitrary bounded open Lipschitz sets $Ω\subset\mathbb{R}^d$, without assuming convexity.
We obtain estimates on the first-order Malliavin derivative of mild solutions, evaluated at fixed points in time and space, to a class of parabolic dissipative stochastic PDEs on bounded domain of $\mathbb{R}^d$. In particular, such equations are driven by multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a locally Lipschitz continuous function satisfying suitable polynomial growth bounds. The main arguments rely on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces, monotonicity, and a comparison p…
Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball
本文的目的是证明球中薛定谔算子第一特征值的一个定量不等式。更准确地说，我们优化的第一特征值$λ(V)$的运营商$\mathcal L_v:=-Δ-V$与Dirichlet边界条件相对于潜在的$V$，$L^1$和$L^\infty$约束下$V$。该解决方案已被称为是一个中心球的特征函数，但本文旨在证明以下形式的急剧增长率：如果$V^*$是一个极小，然后$λ(V)-λ(V^*)\geq C ||V-V^*||_{L^1(Ω)}^2$的一些$C>0$。证明依赖于两个概念的衍生物形状优化：参数衍生物和形状衍生物。我们使用参数导数来处理径向竞争对手，形状导数来处理球的法向变形。一个二分法，然后建立推广的结果，所有其他的潜力。我们发展了一种新的方法来处理径向分布和比较原则来处理二阶形状导数在球。最后，我们添加了一些关于在这种情况下的二阶形状导数的davity范数的备注。
Derivative estimates for SDEs with singular and unbounded coefficients
Anhui University of Science and Technology（安徽理工大学）；Jiangsu Normal University（江苏师范大学）；Beijing Institute of Technology（北京理工大学）；Shenzhen MSU-BIT University（深圳北理莫斯科大学）
The purpose of this note is to provide a summary of the recent work of the authors on two variations of the pointwise convergence problem for the solutions to the fractional Schrödinger equations; convergence along a tangential line and along a set of lines, as exhibiting some new results in each setting. For the former case, we make a simple observation on a path along a tangential curve of exponential order. We discuss counterexamples for the latter case that show some of the known smooth regularities are essentially optimal.
