Related papers: Extremal Functions for the Singular Moser-Trudinge…

We establish some existence results for a class of critical elliptic problems with singular exponential nonlinearities. We do not assume any global sign conditions on the nonlinearity, which makes our results new even in the nonsingular…

Though Trudinger-Moser inequalities on compact Riemannian manifolds or Euclidean space are well understood, we know little about them on complete noncompact Riemannian manifolds. In this paper, we established respectively necessary…

We solve two continuous extremal problems on the classes of monotone functions: in the first problem we find extremal values for a line integral of a coordinate-wise monotone function of two variables from a rearrange\-ment-invariant class…

For $\alpha\geq 2$, we investigate a class of Fourier extension operators on fractional surfaces $(\xi,|\xi|^\alpha)$. For the corresponding $\alpha$-Strichartz inequalities, by applying the missing mass method and bilinear restriction…

The aim of this paper is twofold. In the first part we focus on a functional involving a weighted curvature integral and the quermassintegrals. We prove upper and lower bounds for this functional in the class of convex sets, which provide a…

Let $W^{1,n} ( \mathbb{R}^{n} $ be the standard Sobolev space and $\left\Vert \cdot\right\Vert _{n}$ be the $L^{n}$ norm on $\mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm \[…

Capillarity functionals are parameter invariant functionals defined on classes of two-dimensionals parametric surfaces in R3 as the sum of the area integral with an anisotropic term of suitable form. In the class of parametric surfaces with…

Using the framework first presented by Ruf and Sani, we give a proof of an Adams type inequality which can be applied to a nonlinear functional. Under two kinds of assumptions on the nonlinearity, we estimate the min-max level of the…

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases these inequalities are tight in certain limits. As a consequence, we deduce a tight double…

In this paper, we first show that there exists a maximizer for the non-endpoint Strichartz inequalities for the Schr\"odinger equation in all dimensions based on the recent linear profile decomposition results. We then present a new proof…

We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal…

We prove existence of extremal functions for some Rellich-Sobolev type inequalities involving the $L^{2}$ norm of the Laplacian as a leading term and the $L^{2}$ norm of the gradient, weighted with a Hardy potential. Moreover we exhibit a…

Let us consider the following Caffarelli-Kohn-Nirenberg type inequality \begin{equation}\label{nsckn} \int_{\mathbb{R}^N}|x|^{-\beta}|\mathrm{div} (|x|^{\alpha}\nabla u)|^2 \mathrm{d}x \geq \mathcal{S}\left(\int_{\mathbb{R}^N}|x|^{\gamma}…

If a pair of functions nearly extremizes Young's convolution inequality for R^d, with all three exponents finite and strictly greater than 1, then each function is close in norm to a Gaussian. The proof relies on the Riesz-Sobolev…

We consider the existence and uniqueness of a minimizer of the extremal problem for weighted combined energy between two concentric annuli and obtain that the extremal mapping is a certain radial mapping. Meanwhile, this in turn implies a…

In this paper, we are interested in several questions raised mainly in [17]. We consider the perturbed Moser-Trudinger inequality $I\_\alpha^g(\Omega)$ below, at the critical level $\alpha=4\pi$, where $g$, satisfying $g(t)\to 0$ as $t\to…

We show that complex geometric features of Teichmuller spaces create explicitly the extremals of generic homogeneous holomorphic functionals on univalent functions. In particular this gives proofs of the well-known Zalcman and Bieberbach…

Let $M$ be a complete, simply connected Riemannian manifold with negative curvature. We obtain some Moser-Trudinger inequalities with sharp constants on $M$.

We give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in $1$-dimensional functional inequalities.

We compute the extremal plurisubharmonic function of the real torus viewed as a compact subset of its natural algebraic complexification.