arXiv subjects
Publications and source records attributed to Cyril Tintarev.
The paper studies existence of ground states for the nonlinear Schrödinger equation with a general external magnetic field. In particular, no lattice periodicity or symmetry of the magnetic field, or presence of external electric field is required.
It is common that a Sobolev space defined on $\mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $\mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.
The paper studies existence of solutions for the nonlinear Schrödinger equation with a general bounded external magnetic field. In particular, no lattice periodicity of the magnetic field or presence of external electric field is required. Solutions are obtained by means of a general structural statement about bounded sequences in the magnetic Sobolev space.
For many known non-compact embeddings of two Banach spaces $E\hookrightarrow F$, every bounded sequence in $E$ has a subsequence that takes form of a profile decomposition - a sum of clearly structured terms with asymptotically disjoint supports plus a remainder that vanishes in the norm of $F$. In this note we construct a profile decomposition for arbitrary sequences in the Sobolev space $H^{1,2}(M)$ of a compact Riemannian manifold, relative to the embedding of $H^{1,2}(M)$ into $L^{2^*}(M)$, generalizing the well-known profile decomposition of Struwe ([Proposition 2.1]{Struwe}) to the case of arbitrary bounded sequences.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on manifolds different from M, that are induced by a limiting procedure. The profiles satisfy an inequality of Plancherel type, and a similar relation, related to the Brezis-Lieb Lemma, holds for Lebesgue norms of profiles on the respective manifolds.
The paper studies compactness properties of the affine Sobolev inequality of Gaoyong Zhang et al in the case $p=2$, and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems \[ -\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial x_j}=f \mbox{ in }Ω\subset\mathbb R^N, \] and \[ -\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\frac{\partial^2u}{\partial x_i\partial x_j}=u^{q-1}\,,\quad u>0,\mbox{ in }Ω\subset\mathbb R^N, \] where $A_{ij}[u]=\int_Ω\frac{\partial u}{\partial x_i}\frac{\partial u}{\partial x_j}\mathrm{d}x$ and $q\in(2,\frac{2N}{N-2})$.
We study the existence of ground and bound state solutions for a system of coupled Schrödinger equations with linear and nonlinear couplings in $\mathbb{R}^N$. By studying the limit system and using concentration compactness arguments, we prove the existence of ground and bound state solutions under suitable assumptions. Our results are new even for the limit system.
In this article, we have derived the following Leray-Trudinger type inequality on a bounded domain $Ω$ in $\mathbb{R}^n $ containing the origin. \begin{align*} \displaystyle{\sup_{u\in W^{1,n}_{0}(Ω), I_{n}[u,Ω,R]\leq 1}}\int_Ω e^{c_n\left(\frac{|u(x)|}{E_{2}^β(\frac{|x|}{R})}\right)^{\frac{n}{n-1}}} dx < +\infty \ \text{, for some } c_n>0 \ \text{depending only on } n. \end{align*} Here $β= \frac{2}{n}$, $I_n[u,Ω,R] := \int_Ω|\nabla u |^{n}dx- \left(\frac{n-1}{n}\right)^{n}\int_Ω\frac{|u|^{n}}{|x|^{n}E_{1}^n(\frac{|x|}{R})}dx $, $R \geq \displaystyle{\sup_{x\in Ω}}|x|$ and $E_{1}(t) := \log(\frac{e}{t})$, $E_{2}(t) := \log(eE_1(t))$ for $t\in (0,1].$ This improves an earlier result by Psaradakis and Spector. Also we have proved that, for any $c>0$ the above inequality is false, if we take $β< \frac{1}{n}.$
Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured "bubble decomposition" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis.
We discuss some basic properties of polar convergence in metric spaces. Polar convergence is closely connected with the notion of Delta-convergence of T.C. Lim known for several years. Possible existence of a topology which induces polar convergence is also investigated. Some applications of polar convergence follow.
The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the classical radial estimate. Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity.
The concept of a profile decomposition formalizes concentration compactness arguments on the functional-analytic level, providing a powerful refinement of the Banach-Alaoglu weak-star compactness theorem. We prove existence of profile decompositions for general bounded sequences in uniformly convex Banach spaces equipped with a group of bijective isometries, thus generalizing analogous results previously obtained for Sobolev spaces and for Hilbert spaces. Profile decompositions in uniformly convex Banach spaces are based on the notion of $Δ$-convergence by T. C. Lim instead of weak convergence, and the two modes coincide if and only if the norm satisfies the well-known Opial condition, in particular, in Hilbert spaces and $\ell^{p}$-spaces, but not in $L^{p}(\mathbb R^{N})$, $p\neq2$. $Δ$-convergence appears naturally in the context of fixed point theory for non-expansive maps. The paper also studies connection of $Δ$-convergence with Brezis-Lieb Lemma and gives a version of the latter without an assumption of convergence a.e.
Defect of compactness for non-compact imbeddings of Banach spaces can be expressed in the form of a profile decomposition. This paper extends the profile decomposition for Sobolev spaces proved by Solimini (AIHP 1995) to the non-reflexive case p=1. Since existence of concentration profiles relies on weak-star compactness, the corresponding result is set in a larger, conjugate, space of functions of bounded variation. We prove existence of minimizers for related inequalities and generalizations for to spaces of bounded variation on Lie groups.
Brezis-Lieb lemma is a refinement of Fatou lemma providing an evaluation of the gap between the integral for a sequence and the integral for its pointwise limit. This note studies the question if such gap can be evaluated when there is no a.e. convergence. In particular, it gives the same lower bound for the gap in L^p as the gap in the Brezis-Lieb lemma (including the case vector-valued functions) provided that p is greater or equal than 3 and the sequence converges both weakly and weakly in the sense of a duality map. It also shows that the statement is false if p<3. An application is given in form of a Brezis-Lieb lemma for gradients.
We prove the existence of a sign-changing eigenfunction at the second minimax level of the eigenvalue problem for the scalar field equation under a slow decay condition on the potential near infinity. The proof involves constructing a set consisting of sign-changing functions that is dual to the second minimax class. We also obtain a nonradial sign-changing eigenfunction at this level when the potential is radial.
Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions.
The paper gives an improvement of the Trudinger-Moser inequality, in which the constraint set is defined not by the squared gradient norm, but with the squared gradient norm minus a remainder term of the weighted L^p-type. This is a two-dimensional counterpart of the Hardy-Sobolev-Mazya inequality in higher dimensions, which is a similar refinement of the limiting Sobolev inequality. In particular, we generalize two known cases of remainder terms of potential type (i.e. weighted L^2-terms) found by Adimurthi and Druet and by Wang and Ye. In addition, we prove the inequality with a L^p-remainder, p>2, as well as give an analogous improvement for the Onofri-Beckner inequality.