Global unique continuation from a half space for the Schrödinger equation

We obtain a global unique continuation result for the differential inequality |(i_∂t+Δ)u|?|V(x)u|$|(i{\partial }_t+\mathrm{\Delta })u|?|V(x)u|$ in Rⁿ⁺¹${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger equation with time-independent potentials V(x)$V(x)$ in Rⁿ${\mathbb{R}}^n$. Our method is based on a new type of Carleman estimates for the operator i_∂t+Δ$i{\partial }_t+\mathrm{\Delta }$ on Rⁿ⁺¹${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.     

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$\Omega =B_1$.     

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)$L^2(0,T;H)$.     

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.     

Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function _ct$c_t$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a   and volatility σ>0$\sigma >0$. If β?2a/σ²-1>0$\beta ?2a/{\sigma }^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u)$\Psi (u)$ as the initial endowment u   tends to infinity, i.e. we show that _C*u^-β?Ψ(u)?C^*u^-β$C_{*}{u}^{-\beta }?\Psi (u)?C^{*}{u}^{-\beta }$ for sufficiently large u  . Moreover if _ct=c^*e^γt$c_t=c^{*}{\mathrm{e}}^{\gamma t}$ with γ?0$\gamma ?0$ we find the exact asymptotics of the ruin probability, namely Ψ(u)∼u^-β$\Psi (u)\sim u^{-\beta }$. If β?0$\beta ?0$, we show that Ψ(u)=1$\Psi (u)=1$ for any u?0$u?0$.     

Existence and separation of positive radial solutions for semilinear elliptic equations

We consider the semilinear elliptic equation Δu+K(|x|)u^p=0$\mathrm{\Delta }u+K(|x|)u^p=0$ in R^N${\mathbf{R}}^N$ for N>2$N>2$ and p>1$p>1$, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r^−?K(r)$r^{-?}K(r)$ with ?>−2$?>-2$ around a positive constant is small near r=∞$r=\infty$ and p   is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent _pc$p_c$ which is determined by N   and the order of the behavior of K(r)$K(r)$ as r=|x|→0$r=|x|\to 0$ and ∞. In order to understand how subtle the structure is on K   at p=_pc$p=p_c$, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2)$p=(N+2)/(N-2)$.     

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.     

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$\Omega$.     

Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type

We establish symmetrization results for the solutions of the linear fractional diffusion equation _∂tu+(−Δ)^σ/2u=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}u=f$ and its elliptic counterpart hv+(−Δ)^σ/2v=f$hv+{(-\mathrm{\Delta })}^{\sigma /2}v=f$, h>0$h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, _∂tu+(−Δ)^σ/2A(u)=f${\partial }_{t}u+{(-\mathrm{\Delta })}^{\sigma /2}A(u)=f$, but only when the nondecreasing function A:_R+→_R+$A:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is concave. In the elliptic case, complete symmetrization results are proved for B(v)+(−Δ)^σ/2v=f$B(v)+{(-\mathrm{\Delta })}^{\sigma /2}v=f$ when B(v)$B(v)$ is a convex nonnegative function for v>0$v>0$ with B(0)=0$B(0)=0$, and partial results hold when B is concave. Remarkable counterexamples are constructed for the parabolic equation when A is convex, resp. for the elliptic equation when B   is concave. Such counterexamples do not exist in the standard diffusion case σ=2$\sigma =2$.     

Multiplicity of positive and nodal solutions for scalar field equations

In this paper the question of finding infinitely many solutions to the problem −Δu+a(x)u=|u|^p−2u$-\mathrm{\Delta }u+a(x)u={|u|}^{p-2}u$, in R^N${\mathbb{R}}^N$, u∈H¹(R^N)$u\in H^1({\mathbb{R}}^N)$, is considered when N≥2$N\ge 2$, p∈(2,2N/(N−2))$p\in (2,2N/(N-2))$, and the potential a(x)$a(x)$ is a positive function which is not required to enjoy symmetry properties. Assuming that a(x)$a(x)$ satisfies a suitable “slow decay at infinity” condition and, moreover, that its graph has some “dips”, we prove that the problem admits either infinitely many nodal solutions or infinitely many constant sign solutions. The proof method is purely variational and allows to describe the shape of the solutions.     

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.