相关论文: Optimal order finite difference approximation of g…
Hybrid High-Order methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(\Omega)$. In this paper we devise and analyze a variant thereof, which is defined for any load in the dual…
This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in…
In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the…
We consider the numerical solution of the fractional Laplacian of index $s\in(1/2,1)$ in a bounded domain $\Omega$ with homogeneous boundary conditions. Its solution a priori belongs to the fractional order Sobolev space ${\widetilde…
We investigate the validity of the optimal higher-order Sobolev inequality $H_k^2(M^n)\hookrightarrow L^{\frac{2n}{n-2k}}(M^n)$ on a closed Riemannian manifold when the remainder term is the $L^2-$norm. Unlike the case $k=1$, the optimal…
The paper is concerned with a free boundary problem generated by the biharmonic operator and an obstacle. The main goal is to deduce a fully guaranteed upper bound of the difference between the exact minimizer u and any function…
We find the optimal function norm on the left-hand side of the $m$th order Sobolev type inequality $\|u\|_{Y(\mathbb{H}^n)} \leq C \|\nabla_g^m u\|_{X(\mathbb{H}^n)}$ in the $n$-dimensional hyperbolic space $\mathbb{H}^n$, $1\leq m < n$.…
In this paper, we study the fundamental open question of finding the optimal high-order algorithm for solving smooth convex minimization problems. Arjevani et al. (2019) established the lower bound $\Omega\left(\epsilon^{-2/(3p+1)}\right)$…
Recently, Dolbeault-Esteban-Figalli-Frank-Loss [20] established the optimal stability of the first-order $L^2$-Sobolev inequality with dimension-dependent constant. Subsequently, Chen-Lu-Tang [18] obtained the optimal stability for the…
The compact fourth-order finite-difference scheme for solving the 1d wave equation is studied. New error bounds of the fractional order $\mathcal{O}(h^{4(\lambda-1)/5})$ are proved in the mesh energy norm in terms of data, for two initial…
In this article, we improve the convergence order of some finite volume solutions approximating some second order elliptic problems. We prove that finite volume approximations of order $O(h^{k+1})$, with $k$ integer, can be obtained after…
In this paper, we study optimal quadrature errors, approximation numbers, and sampling numbers in $L_2(\Bbb S^d)$ for Sobolev spaces ${\rm H}^{\alpha,\beta}(\Bbb S^d)$ with logarithmic perturbation on the unit sphere $\Bbb S^d$ in $\Bbb…
A linear implicit finite difference method is proposed for the approximation of the solution to a periodic, initial value problem for a Schrodinger-Hirota equation. Optimal, second order convergence in the discrete $H^1-$norm is proved,…
In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $\Omega \subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in…
Given any finite direction set $\Omega$ of cardinality $N$ in Euclidean space, we consider the maximal directional Hilbert transform $H_{\Omega}$ associated to this direction set. Our main result provides an essentially sharp uniform bound,…
We consider the inhomogeneous biharmonic nonlinear Schr\"odinger equation $$ i u_t +\Delta^2 u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and $\alpha$, $b>0$. In the subctritical case, we improve the global well-posedness…
We study the approximation properties of a harmonic function $u \in H\sp{1-k}(\Omega)$, $k > 0$, on relatively compact sub-domain $A$ of $\Omega$, using the Generalized Finite Element Method. For smooth, bounded domains $\Omega$, we obtain…
We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega,…
We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s<1$, $p=1,2$, $1\leq…
A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based…