Best constants in a borderline case of second-order Moser type inequalities

We study optimal embeddings for the space of functions whose Laplacian Δu   belongs to L¹(Ω)$L^1(\Omega )$, where Ω⊂RN$\Omega \subset {\mathbb{R}}^N$ is a bounded domain. This function space turns out to be strictly larger than the Sobolev space W2,1(Ω)$W^{2,1}(\Omega )$ in which the whole set of second-order derivatives is considered. In particular, in the limiting Sobolev case, when N=2$N=2$, we establish a sharp embedding inequality into the Zygmund space Lexp(Ω)$L_{\mathit{exp}}(\Omega )$. On one hand, this result enables us to improve the Brezis–Merle (Brezis and Merle (1991) [13]) regularity estimate for the Dirichlet problem Δu=f(x)∈L¹(Ω)$\mathrm{\Delta }u=f(x)\in L^1(\Omega )$, u=0$u=0$ on ∂Ω; on the other hand, it represents a borderline case of D.R. Adams' (1988) [1] generalization of Trudinger–Moser type inequalities to the case of higher-order derivatives. Extensions to dimension N?3$N?3$ are also given. Besides, we show how the best constants in the embedding inequalities change under different boundary conditions.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue _μ1(Ω)${\mu }_1(\Omega )$ in Gauss space, where Ω   is a possibly unbounded domain of R^N${\mathbb{R}}^N$. Our main result consists in showing that among all sets Ω   of R^N${\mathbb{R}}^N$ symmetric about the origin, having prescribed Gaussian measure, _μ1(Ω)${\mu }_1(\Omega )$ is maximum if and only if Ω is the Euclidean ball centered at the origin.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

On the radius of spatial analyticity for semilinear symmetric hyperbolic systems

We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (= width) _ε0${\epsilon }_0$, the same happens for the solution u(t,⋅)$u(t,\cdot )$ for a certain radius ε(t)$\epsilon (t)$, as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t)$\epsilon (t)$ as t grows.     

Global solutions with shock waves to the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws

It is proven that the generalized Riemann problem for a class of quasilinear hyperbolic systems of balance laws admits a unique global piecewise C¹$C^1$ solution u=u(t,x)$u=u(t,x)$ containing only n$n$ shock waves with small amplitude on t?0$t?0$ and this solution possesses a global structure similar to that of the similarity solution u=U(x/t)$u=U(x/t)$ of the corresponding homogeneous Riemann problem. As an application of our result, we prove the existence of global shock solutions, piecewise continuous and piecewise smooth solution with shock discontinuities, of the flow equations of a model class of fluids with viscosity induced by fading memory with a single jump initial data.     

Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$: If a Leray–Hopf weak solution is Hölder continuous θ∈Cδ(R²)$\theta \in C^{\delta }({\mathbb{R}}^2)$ with δ>1−2α$\delta >1-2\alpha$ on the time interval [t₀,t]$[t_0,t]$, then it is actually a classical solution on (t₀,t]$(t_0,t]$.     

An improvement for the Trudinger–Moser inequality and applications

In line with the Concentration–Compactness Principle due to P.-L. Lions [19], we study the lack of compactness of Sobolev embedding of W^1,n(Rⁿ)$W^{1,n}({\mathbb{R}}^n)$, n?2$n?2$, into the Orlicz space _LΦα$L_{{\Phi }_{\alpha }}$ determined by the Young function _Φα(s)${\Phi }_{\alpha }(s)$ behaving like e^{α|s|n/(n−1)}−1$e^{\alpha {|s|}^{n/(n-1)}}-1$ as |s|→+∞$|s|\to +\infty$. In the light of this result we also study existence of ground state solutions for a class of quasilinear elliptic problems involving critical growth of the Trudinger–Moser type in the whole space Rⁿ${\mathbb{R}}^n$.     

An approximation method for strictly pseudocontractive mappings

Let K$K$ be a closed convex subset of a q$q$-uniformly smooth separable Banach space, T:K→K$T:K\to K$ a strictly pseudocontractive mapping, and f:K→K$f:K\to K$ an L$L$-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1)$t\in (0,1)$, let _xt$x_t$ be the unique fixed point of tf+(1-t)T$\mathit{tf}+(1-t)T$. We prove that if T$T$ has a fixed point, then {_xt}$\{x_t\}$ converges to a fixed point of T$T$ as t$t$ approaches to 0.     

Capturing two elements in unavoidable minors of 3-connected binary matroids

Let M be a 3-connected binary matroid and let n   be an integer exceeding 2. Ding, Oporowski, Oxley, and Vertigan proved that there is an integer f(n)$f(n)$ so that if |E(M)|>f(n)$|E(M)|>f(n)$, then M has a minor isomorphic to one of the rank-n wheel, the rank-n   tipless binary spike, or the cycle or bond matroid of _K3,n$K_{3,n}$. This result was recently extended by Chun, Oxley, and Whittle to show that there is an integer g(n)$g(n)$ so that if |E(M)|>g(n)$|E(M)|>g(n)$ and x∈E(M)$x\in E(M)$, then x is an element of a minor of M isomorphic to one of the rank-n wheel, the rank-n   binary spike with a tip and a cotip, or the cycle or bond matroid of _K1,1,1,n$K_{1,1,1,n}$. In this paper, we prove that, for each i   in {2,3}$\{2,3\}$, there is an integer _hi(n)$h_i(n)$ so that if |E(M)|>_hi(n)$|E(M)|>h_i(n)$ and Z is an i-element rank-2 subset of M, then M has a minor from the last list whose ground set contains Z.     

Laguerre functions on symmetric cones and recursion relations in the real case

In this article we derive differential recursion relations for the Laguerre functions on the cone Ω$\Omega$ of positive definite real matrices. The highest weight representations of the group Sp(n,R)$\mathrm{Sp}(n,\mathbb{R})$ play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω+iSym(n,R)$\Omega +i\mathrm{Sym}(n,\mathbb{R})$. We then use the Laplace transform to carry the Lie algebra action over to L²(Ω,d_μν)$L^2(\Omega ,\mathrm{d}{\mu }_{\nu })$. The differential recursion relations result by restricting to a distinguished three-dimensional subalgebra, which is isomorphic to sl(2,R).$\mathfrak{sl}(2,\mathbb{R}).$