Critical Embedding Research Articles

Existence and uniqueness of two dimensional Euler-Poisson system and WKB approximation to the nonlinear Schrödinger-Poisson system

In this paper, we study a dispersive Euler-Poisson system in two dimensional Euclidean space. Our aim is to show unique existence and the zero-dispersion limit of the time-local weak solution. Since one may not use dispersive structure in the zero-dispersion limit, when reducing the regularity, lack of critical embedding H1⊊L∞ becomes a bottleneck. We hence employ an estimate on the best constant of the Gagliardo-Nirenberg inequality. By this argument, a reasonable convergence rate for the zero-dispersion limit is deduced with a slight loss. We also consider the semiclassical limit problem of the Schrödinger-Poisson system in two dimensions.

Conic D-branes

The shape of D-branes is of fundamental interest in string theory. We find that generically D-branes in trivial spacetime can form a conic shape under external uniform forces. Surprisingly, the apex angle is found to be unique, once the spatial dimensions of the cone is given. In particular it is universal irrespective of the external forces. The quantized angle is reminiscent of Taylor cones of hydrodynamic electrospray. We provide explicit D-brane solutions as well as the mechanism of a force balance on the cone, for D-branes in RR and NSNS flux backgrounds. Critical embedding of probe D-branes in AdS/CFT with electric and magnetic fields is in the same category, for which we give an analytic proof of a power-low spectrum of "turbulent meson condensation."

Trudinger–Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to L^{\infty} . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrödinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

Discrete Dirac Operators, Critical Embeddings and Ihara-Selberg Functions

The aim of the paper is to formulate a discrete analogue of the claim made by Alvarez-Gaume et al., realizing the partition function of the free fermion on a closed Riemann surface of genus $g$ as a linear combination of $2^{2g}$ Pfaffians of Dirac operators. Let $G=(V,E)$ be a finite graph embedded in a closed Riemann surface $X$ of genus $g$, $x_e$ the collection of independent variables associated with each edge $e$ of $G$ (collected in one vector variable $x$) and $\S$ the set of all $2^{2g}$ spin-structures on $X$. We introduce $2^{2g}$ rotations $rot_s$ and $(2|E|\times 2|E|)$ matrices $\Delta(s)(x)$, $s\in \Sigma$, of the transitions between the oriented edges of $G$ determined by rotations $rot_s$. We show that the generating function of the even sets of edges of $G$, i.e., the Ising partition function, is a linear combination of the square roots of $2^{2g}$ Ihara-Selberg functions $I(\Delta(s)(x))$ also called Feynman functions. By a result of Foata and Zeilberger $I(\Delta(s)(x))=\det(I-\Delta'(s)(x))$, where $\Delta'(s)(x)$ is obtained from $\Delta(s)(x)$ by replacing some entries by $0$. Thus each Feynman function is computable in a polynomial time. We suggest that in the case of critical embedding of a bipartite graph $G$, the Feynman functions provide suitable discrete analogues for the Pfaffians of Dirac operators.

Characterization of the lack of compactness of $H^2_{\rm rad}({\mathbb R}^4)$ into the Orlicz space

This paper is devoted to the description of the lack of compactness of the Sobolev space [Formula: see text] in the Orlicz space [Formula: see text]. The approach that we adopt to establish this characterization is in the spirit of the one adopted in the case of [Formula: see text] into the Orlicz space [Formula: see text] in [H. Bahouri, M. Majdoub and N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal. 260 (2011) 208–252].

On the p-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the p-biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

Existence result for a class of biharmonic equations with critical growth and singular potential inRN

In this paper, we prove a compactness theorem for a critical Sobolev embedding involving the biharmonic operator with a singular potential. Then, using the compactness theorem and the Mountain Pass Theorem, we obtain the existence of nontrivial solutions for a class of biharmonic equations with critical growth and a singular potential in R N .

Lack of compactness in the 2D critical Sobolev embedding, the general case

This paper is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2) in the critical Orlicz space L(R2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in Moser (1971) [44] as in the radial setting investigated in Bahouri et al. (2011) [12]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L∞ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of H1(R2), the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.

Lack of compactness in the 2D critical Sobolev embedding, the general case

This Note is devoted to the description of the lack of compactness of the Sobolev embedding of H1(R2) in the critical Orlicz space L(R2). It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser (1971) in [16] as in the radial setting investigated in Bahouri et al. (2011) [5]. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an L∞ estimate far away from the origin and which is no longer valid in the general frame work. The strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations.

A GENERAL WAVELET-BASED PROFILE DECOMPOSITION IN THE CRITICAL EMBEDDING OF FUNCTION SPACES

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.

A proof of Hessian Sobolev inequality

In this paper, taking the Hessian Sobolev inequality ( 0 < p ≤ k ) (X.-J. Wang, 1994 [2]) as the starting point, we give a proof of the Hessian Sobolev inequality when k < p ≤ k ∗ , where k ∗ is the critical Sobolev embedding index of k -Hessian type. We also prove that k ∗ is optimal by one-dimensional Hardy’s inequality.

Description of the lack of compactness of some critical Sobolev embedding

Description of the lack of compactness of some critical Sobolev embedding

On the lack of compactness in the 2D critical Sobolev embedding

This paper is devoted to the description of the lack of compactness of H rad 1 ( R 2 ) in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P.-L. Lions (1985) in [24]. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and takes into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of nonlinear wave equation with exponential growth, where the Orlicz norm plays a decisive role.

EXISTENCE AND NON-EXISTENCE FOR SCHRÖDINGER EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENTS

We study existence of positive solutions of the classical nonlinear Schr<TEX>$\ddot{o}$</TEX>dinger equation <TEX>$-{\Delta}u\;+\;V(x)u\;-\;f(x,\;u)\;-\;H(x)u^{2*-1}\;=\;0$</TEX>, u > 0 in <TEX>$\mathbb{R}^n$</TEX> <TEX>$u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$</TEX>. In fact, we consider the following more general quasi-linear Schr<TEX>$\ddot{o}$</TEX>odinger equation <TEX>$-div(|{\nabla}u|^{m-2}{\nabla}u)\;+\;V(x)u^{m-1}$</TEX> <TEX>$-f(x,\;u)\;-\;H(x)u^{m^*-1}\;=\;0$</TEX>, u > 0 in <TEX>$\mathbb{R}^n$</TEX> <TEX>$u\;{\rightarrow}\;0\;as\;|x|\;{\rightarrow}\;{\infty}$</TEX>, where m <TEX>$\in$</TEX> (1, n) is a positive number and <TEX>$m^*\;:=\;\frac{mn}{n-m}\;</TEX><TEX>></TEX><TEX>\;0$</TEX>, is the corresponding critical Sobolev embedding number in <TEX>$\mathbb{R}^n$</TEX>. Under appropriate conditions on the functions V(x), f(x, u) and H(x), existence and non-existence results of positive solutions have been established.

On the existence of an extremal function in critical Sobolev trace embedding theorem

Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev–Poincaré inequality are established. It is shown that some of these conditions are sharp.

Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions

We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti (Ann Scuola Norm Sup Pisa Cl Sci 3(4): 697–718, 1976) and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for Hardy-Rellich inequalities.

Critical exponents from AdS/CFT with flavor

We use the AdS/CFT correspondence to study the thermodynamics of massive N=2 supersymmetric hypermultiplet flavor fields coupled to N=4 supersymmetric SU(Nc) Yang-Mills theory, formulated on curved four-manifolds, in the limits of large Nc and large 't Hooft coupling. The gravitational duals are probe D-branes in global thermal AdS. These D-branes may undergo a topology-changing transition in the bulk. The D-brane embeddings near the point of the topology change exhibit a scaling symmetry. The associated scaling exponents can be either real- or complex-valued. Which regime applies depends on the dimensionality of a collapsing submanifold in the critical embedding. When the scaling exponents are complex-valued, a first-order transition associated with the flavor fields appears in the dual field theory. Real scaling exponents are expected to be associated with a continuous transition in the dual field theory. For one example with real exponents, the D7-brane, we study the transition in detail. We find two field theory observables that diverge at the critical point, and we compute the associated critical exponents. We also present analytic and numerical evidence that the transition expresses itself in the meson spectrum as a non-analyticity at the critical point. We argue that the transition we study is a true phase transition only when the 't Hooft coupling is strictly infinite.

Best constant in critical Sobolev inequalities of second-order in the presence of symmetries

Let ( M , g ) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H 2 ( M ) ↪ L 2 ♯ ( M ) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of ( M , g ) . We also prove that we can take ϵ = 0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz–Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation.

Best constants in second-order Sobolev inequalities on Riemannian manifolds and applications

Let ( M, g) be a smooth compact Riemannian manifold, with or without boundary, of dimension n⩾3 and 1< p< n/2. Considering the norm ‖u‖= ‖ Δ gu‖ L p(M) p+‖u‖ L p(M) p 1/p on each of the spaces H 2, p ( M), H 0 2, p ( M) and H 2, p ( M)∩ H 0 1, p ( M), we study an asymptotically sharp inequality associated to the critical Sobolev embedding of these spaces. As an application, we investigate the influence of the geometry in the existence of solutions for some fourth-order problems involving critical exponents on manifolds. In particular, new phenomena arise in Brezis–Nirenberg type problems on manifolds with positive scalar curvature somewhere, in contrast with the Euclidean case. We also show that on such manifolds the corresponding optimal inequality for p=2 is not valid.

Tubular phase of self-avoiding anisotropic crystalline membranes.

We analyze the tubular phase of self-avoiding anisotropic crystalline membranes. A careful analysis using renormalization group arguments together with symmetry requirements motivates the simplest form of the large-distance free energy describing fluctuations of tubular configurations. The non-self-avoiding limit of the model is shown to be exactly solvable. For the full self-avoiding model we compute the critical exponents using an epsilon expansion about the upper critical embedding dimension for general internal dimension D and embedding dimension d. We then exhibit various methods for reliably extrapolating to the physical point (D=2,d=3). Our most accurate estimates are nu=0.62 for the Flory exponent and zeta=0.80 for the roughness exponent.