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$\Omega$.     

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]$(t_0,t]$.     

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.     

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.     

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.     

Hölder continuity of solutions of supercritical dissipative hydrodynamic transport equations

We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical (α<1/2$\alpha <1/2$) dissipation α(−Δ)${(-\mathrm{\Delta })}^{\alpha }$. This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical (α=1/2$\alpha =1/2$) QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from L²$L^2$ to L^∞$L^{\infty }$, from L^∞$L^{\infty }$ to Hölder (Cδ$C^{\delta }$, δ>0$\delta >0$), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be L^∞$L^{\infty }$, but it does not appear that their approach can be easily extended to establish the Hölder continuity of L^∞$L^{\infty }$ solutions. In order for their approach to work, we require the velocity to be in the Hölder space C1−2α$C^{1-2\alpha }$. Higher regularity starting from Cδ$C^{\delta }$ with δ>1−2α$\delta >1-2\alpha$ can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press].     

Singular solutions of Hessian elliptic equations in five dimensions

We show that for any δ∈[0,1)$\delta \in [0,1)$ there exists a homogeneous order 2−δ$2-\delta$ analytic outside zero solution to a uniformly elliptic Hessian equation in R⁵${\mathbb{R}}^5$.     

Dynamics of non-autonomous parabolic problems with subcritical and critical nonlinearities

This is a subsequent work of our previous one in [25]. Let the nonlinear terms of non-autonomous parabolic problems with singular initial data satisfy subcritical and critical growth conditions. We first establish the existence of uniform attractors in W^1,r(Ω)$W^{1,r}(\Omega )$, 1<r<N$1<r<N$, for the family of processes corresponding to the equations with external forces being translation bounded but not translation compact. Then, we prove the existence of pullback attractors in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively, for the process corresponding to the equation with the weaker assumption on the external force than previous one. Finally, we investigate the robust of attractors and establish the existence of pullback exponential attractors for the process acting in L^r(Ω)$L^r(\Omega )$ and W^1,r(Ω)$W^{1,r}(\Omega )$, respectively.     

A remark on least-squares mixed element methods for reaction–diffusion problems

In this paper, we propose a least-squares mixed element procedure for a reaction–diffusion problem based on the first-order system. By selecting the least-squares functional properly, the resulting procedure can be split into two independent symmetric positive definite schemes, one of which is for the unknown variable and the other of which is for the unknown flux variable, which lead to the optimal order H¹(Ω)$H^1(\Omega )$ and L²(Ω)$L^2(\Omega )$ norm error estimates for the primal unknown and optimal H(div;Ω)$H(\mathrm{div};\Omega )$ norm error estimate for the unknown flux. Finally, we give some numerical examples.     

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${\mathbb{R}}^n$.     

Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean n space

We study the large time behavior of solutions of the Cauchy problem for the Hamilton–Jacobi equation ut+H(x,Du)=0$u_t+H(x,Du)=0$ in Rn×(0,∞)${\mathbf{R}}^n\times (0,\infty )$, where H(x,p)$H(x,p)$ is continuous on Rn×Rn${\mathbf{R}}^n\times {\mathbf{R}}^n$ and convex in p  . We establish a general convergence result for viscosity solutions u(x,t)$u(x,t)$ of the Cauchy problem as t→∞$t\to \infty$.     

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u   is a solution of (−Δ)^su=g${(-\mathrm{\Delta })}^{s}u=g$ in Ω  , u≡0$u\equiv 0$ in Rⁿ\Ω${\mathbb{R}}^n\backslash \Omega$, for some s∈(0,1)$s\in (0,1)$ and g∈L^∞(Ω)$g\in L^{\infty }(\Omega )$, then u   is C^s(Rⁿ)$C^s({\mathbb{R}}^n)$ and u/δ^s_|Ω$u/{\delta }^s{|}_{\Omega }$ is C^α$C^{\alpha }$ up to the boundary ∂Ω   for some α∈(0,1)$\alpha \in (0,1)$, where δ(x)=dist(x,∂Ω)$\delta (x)=\mathrm{dist}(x,\partial \Omega )$. For this, we develop a fractional analog of the Krylov boundary Harnack method.     

Semilinear fractional elliptic equations involving measures

We study the existence of weak solutions to (E) (−Δ)^αu+g(u)=ν${(-\mathrm{\Delta })}^{\alpha }u+g(u)=\nu$ in a bounded regular domain Ω   in R^N(N≥2)${\mathbb{R}}^N(N\ge 2)$ which vanish in R^N?Ω${\mathbb{R}}^N?\Omega$, where (−Δ)^α${(-\mathrm{\Delta })}^{\alpha }$ denotes the fractional Laplacian with α∈(0,1)$\alpha \in (0,1)$, ν is a Radon measure and g is a nondecreasing function satisfying some extra hypotheses. When g satisfies a subcritical integrability condition, we prove the existence and uniqueness of weak solution for problem (E) for any measure. In the case where ν   is a Dirac measure, we characterize the asymptotic behavior of the solution. When g(r)=|r|^k−1r$g(r)={|r|}^{k-1}r$ with k supercritical, we show that a condition of absolute continuity of the measure with respect to some Bessel capacity is a necessary and sufficient condition in order (E) to be solved.