Related papers: Two-step minimization approach to Sobolev-type ine…
We introduce a family of conformal invariants associated to a smooth metric measure space which generalize the relationship between the Yamabe constant and the best constant for the Sobolev inequality to the best constants for…
We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.
We give a simple proof of the existence of a minimizer for the Sobolev inequality. Our proof is based on a representation formula via a cut-off fundamental solution.
The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $n^{-1/2}$. However, re-weighting of random points can sometimes be used to improve the convergence order.…
In this paper, we establish the sharp fractional subelliptic Sobolev inequalities and Gagliardo-Nirenberg inequalities on stratified Lie groups. The best constants are given in terms of a ground state solution of a fractional subelliptic…
This paper investigates instances of Sobolev embeddings characterized by local compactness at every point within their domain, except for a single point. We obtain the sharp conditions that distinguish compactness from non-compactness and…
This paper is devoted to improvements of functional inequalities based on scalings and written in terms of relative entropies. When scales are taken into account and second moments fixed accordingly, deficit functionals provide explicit…
The Riesz-Sobolev inequality provides an upper bound, in integral form, for the convolution of indicator functions of subsets of Euclidean space. We formulate and prove a sharper form of the inequality. This can be equivalently phrased as a…
In this paper we investigate the existence of solutions to a nonlinear elliptic problem involving critical Sobolev exponent for a polyharmonic operator on a Riemannian manifold $M$. We first show that the best constant of the Sobolev…
We consider a homogeneous space X=(X,d,m) of dimension $\nu \geq 1$ and a local regular Dirichlet form in L^{2}(X,m). We prove that if a Poincar\'{e} inequality holds on every pseudo-ball B(x,R) of X, with local characteristic constant…
We study the Neumann Green's function for second order parabolic systems in divergence form with time-dependent measurable coefficients in a cylindrical domain $\mathcal{Q}=\Omega\times (-\infty,\infty)$, where $\Omega\subset \mathbb{R}^n$…
In the first part of this contribution we prove the global existence and uniqueness of a trajectory that globally converges to the minimizer of the Gross-Pitaevskii energy functional for a large class of external potentials. Using the…
The best constants of two kinds of discrete Sobolev inequalities on the C60 fullerene buckyball are obtained. All the eigenvalues of discrete Laplacian $A$ corresponding to the buckyball are found. They are roots of algebraic equation at…
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to…
In this paper, we analyze a second-order differential equation with a piecewise constant argument and reflection coupled to periodic boundary conditions. Our main contribution is the construction of the related Green's function and a…
In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers.…
We study a class of logarithmic Sobolev inequalities with a general form of the energy functional. The class generalizes various examples of modified logarithmic Sobolev inequalities considered previously in the literature. Refining a…
The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using…
This note presents a method based on Feynman-Kac semigroups for logarithmic Sobolev inequalities. It follows the recent work of Bonnefont and Joulin on intertwining relations for diffusion operators, formerly used for spectral gap…
We present a general formula for the particular solution of an inhomogeneous linear difference equation with variable coefficients. The answer is expressed as a weighted sum of fundamental solutions of the associated linear difference…