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Kyril Tintarev

Publications and source records attributed to Kyril Tintarev.

14 recordsLinked to original sources
arXiv2013

On compactness in the Trudinger-Moser inequality

The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be weakly continuous only in an exceptional case of a concentrating sequence of functions from the Moser family (up to translations and the remainder vanishing in Sobolev norm). The argument is based on a structural description of the defect of weak converegence, analogous to the profile decomposition established by Solimini for Sobolev inequalities, but involving gauge operators specific for the two-dimensional case.

math.AP↗
arXiv2011

On interpolation of cocompact imbeddnings

Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we show that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces over \mathbb{R}^{N} are cocompact relative to lattice shifts. We use techniques of interpolation spaces to deduce our results from known cocompact imbeddings for classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We give examples of applications of cocompactness to compactness of imbeddings of some radial subspaces and to existence of minimizers in some isoperimetric problems.Our research complements a range of previous results, and recalls that there is a natural conceptual framework for unifying them.

math.AP↗
arXiv2011

Ground state for the Schrödinger operater with the weighted Hardy potential

We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

math.AP↗
arXiv2010

Is the Trudinger-Moser nonlinearity a true critical nonlinearity?

While the critical nonlinearity $\int |u|^{2^*}$ for the Sobolev space $H^1$ in dimension $N>2$ lacks weak continuity at any point, Trudinger-Moser nonlinearity $\int e^{4πu^2}$ in dimension $N=2$ is weakly continuous at any point except zero. In the former case the lack of weak continuity can be attributed to invariance with respect to actions of translations and dilations. The Sobolev space $H_0^1$ of the unit disk $\mathbb D\subset\R^2$ possesses transformations analogous to translations (Möbius transformations) and nonlinear dilations $r\mapsto r^s$. We present improvements of the Trudinger-Moser inequality with sharper nonlinearities sharper than $\int e^{4πu^2}$, that lack weak continuity at any point and possess (separately), translation and dilation invariance. We show, however, that no nonlinearity of the form $\int F(|x|,u(x))\mathrm{d}x$ is both dilation- and Möbius shift-invariant. The paper also gives a new, very short proof of the conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and Sandeep and of a sharper version of Onofri-type inequality of Beckner.

math.AP↗
arXiv2010

On the Hardy-Sobolev-Maz'ya inequality and its generalizations

The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.

math.AP↗
arXiv2009

On positive solutions of p-Laplacian-type equations

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by $$Q(u):= \frac{1}{p}\int_Ω. (|\nabla u|^p+V|u|^p) \dx, Q^\prime (u):= -\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2} u.$$ In this paper we discuss a few aspects of relations between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.

math.AP↗
arXiv2008

Hardy inequalities for weighted Dirac operatos

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$ vanishes on a countable set of $b$, which extends the known case $n=2$, $b=0$ which corresponds to the trivial Hardy inequality in $\R^2$. Analogous inequalities are proved in the case $c_b=0$ under constraints and, with error terms, for a bounded domain.

math.AP↗
arXiv2008

Cocompact imbeddings and structure of weakly convergent sequences

Concentration compactness method is a powerful techniques for establishing existence of minimizers for inequalities and of critical points of functionals in general. The paper gives a functional-analytic formulation for the method in Banach space, generalizing the Hilbert space case elaborated in \cite{ccbook}. The key object is a dislocation space - a triple $(X,F,D)$, where $F$ is a convex functional that defines a norm on Banach space $X$, and $D$ is a group of isometries on $X$. Bounded sequences in dislocation spaces admit a decomposition into an asymptotic sum "profiles" $w^{(n)}\in X$ dislocated by actions of $D$, that is, a sum of the form $\sum_ng^{(n)}_kw^{(n)}$, $g^{(n)}_k\in D$, while the remainder term converges weakly under actions of any sequence $g_k\in D$ ({\em $D$-weak convergence}). This decomposition allows to extend the weak convergence argument from variational problems with compactness to problems where $X$ is {\em cocompactly} (relatively to the group $D$) imbedded into a Banach space $Y$, that is, when every sequence $D$-weakly convergent in $X$ is convergent in the norm of $Y$. We prove a general statement on existence of minimizers in cocompact imbeddings that applies, in particular to Sobolev imbeddings which lack compactness (unbounded domain, critical exponent) including the subelliptic Sobolev spaces and spaces over Riemannian manifolds.

math.AP↗
arXiv2007

On positive solutions of minimal growth for singular p-Laplacian with potential term

Let $Ω$ be a domain in $\mathbb{R}^d$, $d\geq 2$, and $1<p<\infty$. Fix $V\in L_{\mathrm{loc}}^\infty(Ω)$. Consider the functional $Q$ and its Gâteaux derivative $Q^\prime$ given by Q(u):=\frac{1}{p}\int_Ω(|\nabla u|^p+V|u|^p)\dx, Q^\prime (u):=-\nabla\cdot(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u. It is assumed that $Q\geq 0$ on $C_0^\infty(Ω)$. In a previous paper we discussed relations between the absence of weak coercivity of the functional $Q$ on $C_0^\infty(Ω)$ and the existence of a generalized ground state. In the present paper we study further relationships between functional-analytic properties of the functional $Q$ and properties of positive solutions of the equation $Q^\prime (u)=0$.

math.AP↗
arXiv2007

A Liouville-type theorem for the p-Laplacian with potential term

In this paper we prove a sufficient condition, in terms of the behavior of a ground state of a singular p-Laplacian problem with a potential term, such that a nonzero subsolution of another such problem is also a ground state. Unlike in the linear case (p=2), this condition involves comparison of both the functions and of their gradients.

math.AP↗
arXiv2005

Concentration-compactness principle for mountain pass problems

In the paper we show that critical sequences associated with the mountain pass level for semilinear elliptic problems on $\R^N$ converge when the non-linearity is subcritical, superlinear and satisfies the penalty condition $F_\infty(s)<F(x,s)$. The proof suggests a concentration compactness framework for minimax problems, similar to that of P.-L.Lions for constrained minima.

math.AP↗
arXiv2004

A ground state alternative for singular Schrödinger operators

Let $\mathbf{a}$ be a quadratic form associated with a Schrödinger operator $L=-\nabla\cdot(A\nabla)+V$ on a domain $Ω\subset \mathbb{R}^d$. If $\mathbf{a}$ is nonnegative on $C_0^{\infty}(Ω)$, then either there is $W>0$ such that $\int W|u|^2 dx\leq \mathbf{a}[u]$ for all $C_0^{\infty}(Ω;\mathbb{R})$, or there is a sequence $ϕ_k\in C_0^{\infty}(Ω)$ and a function $ϕ>0$ satisfying $Lϕ=0$ such that $\mathbf{a}[ϕ_k]\to 0$, $ϕ_k\toϕ$ locally uniformly in $Ω\setminus\{x_0\}$. This dichotomy is equivalent to the dichotomy between $L$ being subcritical resp. critical in $Ω$. In the latter case, one has an inequality of Poincaré type: there exists $W>0$ such that for every $ψ\in C_0^\infty(Ω;\mathbb{R})$ satisfying $\int ψϕdx \neq 0$ there exists a constant $C>0$ such that $C^{-1}\int W|u|^2 dx\le \mathbf{a}[u]+C|\int u ψdx|^2$ for all $u\in C_0^\infty(Ω;\mathbb{R})$.

math.AP↗
arXiv2004

Existence of minimizers for Schrodinger operators under domain perturbations with application to Hardy's inequality

The paper studies the existence of minimizers for Rayleigh quotients $μ_Ω=\inf\frac{\int_Ω|\nabla u|^2}{\int_ΩV{|u|^2}} $, where $Ω$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative function that may have singularities on $\partialΩ$. As a model for our results one can take $Ω$ to be a Lipschitz cone and $V$ to be the Hardy potential $V(x)=\frac{1}{|x|^2} $.

math.AP↗