相关论文: On some nonlinear extensions of the Gagliardo-Nire…
We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in…
We obtain the inequalities of the form $$\int_{\Omega}|\nabla u(x)|^2h(u(x))\,{\rm d} x\leq C\int_{\Omega} \left( \sqrt{ |P u(x)||{\cal T}_{H}(u(x))|}\right)^{2}h(u(x))\,{\rm d} x +\Theta,$$ where $\Omega\subset \mathbf{R}^n$ is a bounded…
We obtain the inequality $$\int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $\Omega\subseteq {\bf R}^n$ and $n\ge 2$, $u:\Omega\rightarrow…
Eigenvalue problems for linear differential equations, such as time-independent Schr\"odinger equations, can be generalized to eigenvalue problems for nonlinear differential equations. In the nonlinear context a separatrix plays the role of…
The classical Gagliardo-Nirenberg inequality, known as an interpolation inequality, involves Lebesgue norms of functions and their derivatives. We established an interpolation lemma to connect Lebesgue and H\"older spaces, thus extending…
This paper is devoted to one-dimensional interpolation Gagliardo-Nirenberg-Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some nonlinear diffusion equations…
We establish Gagliardo-Nirenberg-Sobolev type inequalities on nonlocal Sobolev spaces driven by $p$-L\'{e}vy integrable kernels, by imposing some appropriate growth conditions on the associated critical function. This naturally allows to…
In this paper we prove the Fractional Gagliardo-Nirenberg Inequality, Polya-Szego Inequality and the Sharp Fractional Sobolev Inequality, we then provide an application of such inequalities in a constraiend variational problem involving the…
We establish a Gagliardo-Nirenberg-type inequality in $\mathbb{R}^n$ for functions which decay fast as $|x|\to\infty$. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation.…
Zhang refined the classical Sobolev inequality $\|f\|_{L^{Np/(N-p)}} \lesssim \| \nabla f \|_{L^p}$, where $1\leq p \lt N$, by replacing $\|\nabla f\|_{L^p}$ with a smaller quantity invariant by unimodular affine transformations. The…
In this work, we study the Sobolev inequality on noncommutative Euclidean spaces. As a simple consequence, we obtain the Gagliardo-Nirenberg type inequality and as its application we show global well-posedness of nonlinear PDEs in the…
We introduce and study a new functional which was motivated by our paper on the Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, R\u{a}dulescu and Repov\v{s}, Nonlinearity 31 (2018), 1518-1534). We also study the…
Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems,…
We consider the non-linear eigenvalue equations characterizing $L^p$ into $L^q$ Sobolev embeddings of second order for Navier boundary conditions at both ends of a line segment. We give a complete description of the s-numbers and the…
Building on the inequalities for homogeneous tetrahedral polynomials in independent Gaussian variables due to R. Lata{\l}a we provide a concentration inequality for non-necessarily Lipschitz functions $f\colon \R^n \to \R$ with bounded…
In this paper we give a geometric condition which ensures that $(q,p)$-Poincar\'e-Sobolev inequalities are implied from generalized $(1,1)$-Poincar\'e inequalities related to $L^1$ norms in the context of product spaces. The concept of…
This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine $L^{p}-$Sobolev inequalities. The logarithmic version of affine…
A Hardy inequality of the form \[\int_{\tilde{\Omega}} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tilde{\Omega}} \{1 + a(\delta, \partial \tilde{\Omega})(\x)\}\frac{|f({\bf{x}})|^p}{\delta({\bf{x}})^p} d{\bf{x}}, \] for…
Interpolation inequalities in Triebel-Lizorkin-Lorentz spaces and Besov-Lorentz spaces are studied for both inhomogeneous and homogeneous cases. First we establish interpolation inequalities under quite general assumptions on the parameters…
The paper is concerned with higher order Calderon-Zygmund estimates for the $p$-Laplace equation $$ -\textrm{div}(A(\nabla u)) := -\textrm{div}{(|\nabla u|^{p-2}\nabla u)}=-\textrm{div} F, \qquad 1<p<\infty. $$ We are able to transfer local…