Boundary value problems in spaces of distributions on smooth and polygonal domains

We study boundary value problems of the form -Δu=f$-\mathrm{\Delta }u=f$ on Ω$\Omega$ and Bu=g$\mathit{Bu}=g$ on the boundary ∂Ω$\mathrm{\partial }\Omega$, with either Dirichlet or Neumann boundary conditions, where Ω$\Omega$ is a smooth bounded domain in Rⁿ${\mathbb{R}}^n$ and the data f,g$f,g$ are distributions  . This problem has to be first properly reformulated and, for practical applications, it is of crucial importance to obtain the continuity of the solution u$u$ in terms of f and g  . For f=0$f=0$, taking advantage of the fact that u$u$ is harmonic on Ω$\Omega$, we provide four formulations of this boundary value problem (one using nontangential limits of harmonic functions, one using Green functions, one using the Dirichlet-to-Neumann map, and a variational one); we show that these four formulations are equivalent. We provide a similar analysis for f≠0$f\ne 0$ and discuss the roles of f and g, which turn to be somewhat interchangeable in the low regularity case. The weak formulation is more convenient for numerical approximation, whereas the nontangential limits definition is closer to the intuition and easier to check in concrete situations. We extend the weak formulation to polygonal domains using weighted Sobolev spaces. We also point out some new phenomena for the “concentrated loads” at the vertices in the polygonal case.     

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.     

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$\Omega$.     

Global existence and causality for a hyperbolic transmission problem with a repulsive nonlinearity

It is well known that the solution of the classical linear wave equation with an initial condition with compact support and vanishing initial velocity also has a compact support included in a set depending on time: the support of the solution at time t$t$ is causally related to that of the initial condition. Reed and Simon have shown that for a real-valued Klein–Gordon equation with (nonlinear) right-hand side −λu³$-\lambda u^3$ (λ>0$\lambda >0$), causality still holds. We show the same property for a one-dimensional Klein–Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side F(u)$F\left(u\right)$. We also prove the global existence of a solution using the repulsiveness of F$F$. In the particular case F(u)=−λu³$F\left(u\right)=-\lambda u^3$, the problem is a relativistic model for a quantum particle with repulsive self-interaction and tunnel effect at a semi-infinite potential step.     

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.     

Multiple solutions for a class of semilinear elliptic equations with general potentials

In this paper, we study the existence of infinitely many nontrivial solutions for a class of semilinear elliptic equations −△u+a(x)u=g(x,u)$-△u+a\left(x\right)u=g(x,u)$ in a bounded smooth domain of R^N(N≥3)${\mathbb{R}}^N(N\ge 3)$ with the Dirichlet boundary value, where the primitive of the nonlinearity g$g$ is of superquadratic growth near infinity in u$u$ and the potential a$a$ is allowed to be sign-changing. Recent results in the literature are generalized and significantly improved.     

Critical exponents for non-simultaneous blow-up in a localized parabolic system

The paper deals with the radially symmetric solutions of _ut=Δu+u^m(x,t)vⁿ(0,t)$u_t=\Delta u+u^m(x,t)v^n(0,t)$, _vt=Δv+u^p(0,t)v^q(x,t)$v_t=\Delta v+u^p(0,t)v^q(x,t)$, subject to null Dirichlet boundary conditions. For the blow-up classical solutions, we propose the critical exponents for non-simultaneous blow-up by determining the complete and optimal classification for all the non-negative exponents: (i) There exist initial data such that u$u$ (v$v$) blows up alone if and only if m>p+1$m>p+1$ (q>n+1$q>n+1$), which means that any blow-up is simultaneous if and only if m≤p+1$m\le p+1$, q≤n+1$q\le n+1$. (ii) Any blow-up is u$u$ (v$v$) blowing up with v$v$ (u$u$) remaining bounded if and only if m>p+1$m>p+1$, q≤n+1$q\le n+1$ (m≤p+1$m\le p+1$, q>n+1$q>n+1$). (iii) Both non-simultaneous and simultaneous blow-up may occur if and only if m>p+1$m>p+1$, q>n+1$q>n+1$. Moreover, we consider the blow-up rate and set estimates which were not obtained in the previously known work for the same model.     

On a family of differential operators with the coupling parameter in the boundary condition

We study a family of differential operators _Lα${\mathbf{L}}_{\alpha }$ in two variables, depending on the coupling parameter α?0$\alpha ?0$ that appears only in the boundary conditions. Our main concern is the spectral properties of _Lα${\mathbf{L}}_{\alpha }$, which turn out to be quite different for α<1$\alpha <1$ and for α>1$\alpha >1$. In particular, _Lα${\mathbf{L}}_{\alpha }$ has a unique self-adjoint realization for α<1$\alpha <1$ and many such realizations for α>1$\alpha >1$. In the more difficult case α>1$\alpha >1$ an analysis of non-elliptic pseudodifferential operators in dimension one is involved.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation _ut=Δu+u^p$u_t=\mathrm{\Delta }u+u^p$ on R^N${\mathbb{R}}^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈R^N$x\in {\mathbb{R}}^N$ and t∈R$t\in \mathbb{R}$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0$t?0$, then it necessarily converges to 0, as t→∞$t\to \infty$, uniformly with respect to x∈R^N$x\in {\mathbb{R}}^N$.