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.     

The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation

In this paper we establish a blow up rate of the large positive solutions of the singular boundary value problem -Δu=λu-b(x)u^p,u_|∂Ω=+∞$-\mathrm{\Delta }u=\lambda u-b(x)u^p,u{|}_{\mathrm{\partial }\Omega }=+\infty$ with a ball domain and radially function b(x)$b(x)$. All previous results in the literature assumed the decay rate of b(x)$b(x)$ to be approximated by a distance function near the boundary ∂Ω$\mathrm{\partial }\Omega$. Obtaining the accurate blow up rate of solutions for general b(x)$b(x)$ requires more subtle mathematical analysis of the problem.     

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$\Omega$.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Maximum principles and isoperimetric inequalities for some Monge–Ampère-type problems

In this note we derive a maximum principle for an appropriate functional combination of u(x)$u(\mathbf{x})$ and |∇u|²${|\mathrm{\nabla }u|}^2$, where u(x)$u(\mathbf{x})$ is a strictly convex classical solution to a general class of Monge–Ampère equations. This maximum principle is then employed to establish some isoperimetric inequalities of interest in the theory of surfaces of constant Gauss curvature in R^N+1${\mathbb{R}}^{N+1}$.     

Small time global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation _yt−_yxx+y_yx=u(t)$y_t-y_{xx}+y{y}_x=u(t)$ on the line segment [0,1]$[0,1]$. The right-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1)=0$y(t,1)=0$ and restrict ourselves to using only two controls (namely the interior one u(t)$u(t)$ and the boundary one y(t,0)$y(t,0)$). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole–Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

A semilinear elliptic problem on unbounded domains with reverse penalty

In the well-known work of P.-L. Lions [The concentration–compactness principle in the calculus of variations, The locally compact case, part 1. Ann. Inst. H. Poincaré, Analyse Non Linéaire 1 (1984) 109–1453] existence of positive solutions to the equation -Δu+u=b(x)u^p-1$-\mathrm{\Delta }u+u=b(x)u^{p-1}$, u>0$u>0$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, p∈(2,2N/(N-2))$p\in (2,2N/(N-2))$ was proved under assumption b(x)?_b∞?_lim|x|→∞b(x)$b(x)?b_{\infty }?{\lim }_{\left|x\right|\to \infty }b(x)$. In this paper we prove the existence for certain functions b   satisfying the reverse inequality b(x)<_b∞$b(x)<b_{\infty }$. For any periodic lattice L   in R^N${\mathbb{R}}^N$ and for any b∈C(R^N)$b\in C({\mathbb{R}}^N)$ satisfying b(x)<_b∞$b(x)<b_{\infty }$, _b∞>0$b_{\infty }>0$, there is a finite set Y⊂L$Y\subset L$ and a convex combination b^Y$b^Y$ of b(·-y)$b(·-y)$, y∈Y$y\in Y$, such that the problem -Δu+u=b^Y(x)u^p-1$-\mathrm{\Delta }u+u=b^Y(x)u^{p-1}$ has a positive solution u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$.     

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

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 variational treatment for general elliptic equations of the flame propagation type: regularity of the free boundary

We develop a variational theory to study the free boundary regularity problem for elliptic operators: Lu=Dj(a_ij(x)Diu)+biui+c(x)u=0$Lu=D_j(a_{ij}(x)D_{i}u)+b_i{u}_i+c(x)u=0$ in {u>0}$\{u>0\}$, 〈a_ij(x)∇u,∇u〉=2$〈a_{ij}(x)\mathrm{\nabla }u,\mathrm{\nabla }u〉=2$ on ∂{u>0}$\partial \{u>0\}$. We use a singular perturbation framework to approximate this free boundary problem by regularizing ones of the form: Luε=βε(uε)$L{u}_{\epsilon }={\beta }_{\epsilon }(u_{\epsilon })$, where βε${\beta }_{\epsilon }$ is a suitable approximation of Dirac delta function δ₀${\delta }_0$. A useful variational characterization to solutions of the above approximating problem is established and used to obtain important geometric properties that enable regularity of the free boundary. This theory has been developed in connection to a very recent line of research as an effort to study existence and regularity theory for free boundary problems with gradient dependence upon the penalization.     

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.     

Reduced limit for semilinear boundary value problems with measure data

We study boundary value problems for semilinear elliptic equations of the form −Δu+g°u=μ$-\mathrm{\Delta }u+g°u=\mu$ in a smooth bounded domain Ω⊂R^N$\Omega \subset R^N$. Let {_μn}$\{{\mu }_n\}$ and {_νn}$\{{\nu }_n\}$ be sequences of measure in Ω and ∂Ω   respectively. Assume that there exists a solution _un$u_n$ with data (_μn,_νn)$({\mu }_n,{\nu }_n)$, i.e., _un$u_n$ satisfies the equation with μ=_μn$\mu ={\mu }_n$ and has boundary trace _νn${\nu }_n$. Further assume that the sequences of measures converge in a weak sense to μ and ν   respectively while {_un}$\{u_n\}$ converges to u   in L¹(Ω)$L^1(\Omega )$. In general u   is not a solution of the boundary value problem with data (μ,ν)$(\mu ,\nu )$. However there exists a pair of measures (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ such that u   is a solution of the boundary value problem with this data. The pair (μ^?,ν^?)$({\mu }^{?},{\nu }^{?})$ is called the reduced limit of the sequence {(_μn,_νn)}$\{({\mu }_n,{\nu }_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence. A closely related problem was studied by Marcus and Ponce [3].