Energy decay for the linear Klein–Gordon equation and boundary control

In this work we study the asymptotic behavior of the solutions of the linear Klein–Gordon equation in R^N${\mathbb{R}}^N$, N?1$N?1$. We prove that local energy of solutions to the Cauchy problem decays polynomially. Afterwards, we use the local decay of energy to study exact boundary controllability for the linear Klein–Gordon equation in general bounded domains of R^N${\mathbb{R}}^N$, N?1$N?1$.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

Constant mean curvature surfaces and mean curvature flow with non-zero Neumann boundary conditions on strictly convex domains

In this paper, we study nonparametric surfaces over strictly convex bounded domains in ${\mathbb{R}}^n$, which are evolving by the mean curvature flow with Neumann boundary value. We prove that solutions converge to the ones moving only by translation. And we will prove the existence and uniqueness of the constant mean curvature equation with Neumann boundary value on strictly convex bounded domains.     

Uniqueness of the Cheeger set of a convex body

We prove that if C⊂R^N$C\subset {\mathbb{R}}^N$ is an open bounded convex set, then there is only one Cheeger set inside C$C$ and it is convex. A Cheeger set of C$C$ is a set which minimizes the ratio perimeter over volume among all subsets of C$C$.     

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$x\in {\mathbb{R}}^N$.     

The variational approach to Hamilton–Jacobi equations driven by a Gaussian noise

The global existence and uniqueness of viscosity solutions to the Cauchy problem for the Hamilton–Jacobi equations in R^N${\mathbb{R}}^N$ driven by additive and multiplicative Wiener processes are studied for convex Hamiltonians via variational techniques. The finite speed of propagation is also established in the multiplicative noise case for equations with Lipschitzian Hamiltonians.     

Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities

We extend our recent results on the classification of stable solutions of the equation −Δu=f(u)$-\Delta u=f\left(u\right)$ in R^N${\mathbb{R}}^N$, where f≥0$f\ge 0$ is a general convex, non-decreasing function.     

Infinite time gradient blow-up for parabolic prescribed mean curvature equations

We prove some results on the existence of infinite time gradient blow-up phenomena for parabolic prescribed mean curvature equations over bounded, mean-convex domains in Rⁿ$R^n$.     

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${\mathbb{R}}^{N+1}$.     

Conditions at infinity for the inhomogeneous filtration equation

We investigate existence and uniqueness of solutions to the filtration equation with an inhomogeneous density in R^N${\mathbb{R}}^N$ (N?3$N?3$), approaching at infinity a given continuous datum of Dirichlet type.     

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

A quasiplane f(V)$f(V)$ is the image of an n-dimensional Euclidean subspace V   of R^N${\mathbb{R}}^N$ (1≤n≤N−1$1\le n\le N-1$) under a quasiconformal map f:R^N→R^N$f:{\mathbb{R}}^N\to {\mathbb{R}}^N$. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of Rⁿ${\mathbb{R}}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−n$N-n$. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.     

Non-convex CMC spheres

In this paper, we will construct examples of non-convex constant mean curvature spheres in R³${\mathbb{R}}^3$ with a local perturbation of the Euclidean metric. When the perturbation area becomes small and converges to the origin, the non-convex constant mean curvature sphere converges in the Hausdorff sense to the singular space of two tangent unit spheres.     

Remarks on multiplicity of solutions for a subquadratic elliptic equation

We consider a subquadratic elliptic equation in a bounded domain Ω⊂R^N$\Omega \subset {\mathbb{R}}^N$ (N?1$N?1$). By the Clark Theorem, we obtain the existence and multiplicity of its nontrivial solutions, and we show that this result has a great relationship with Ω itself. The above argument can be extended to biharmonic equations.     

Estimates for eigenvalues of the Paneitz operator

For an n  -dimensional compact submanifold Mⁿ$M^n$ in the Euclidean space R^N${\mathbf{R}}^N$, we study estimates for eigenvalues of the Paneitz operator on Mⁿ$M^n$. Our estimates for eigenvalues are sharp.     

The topological structure of fuzzy sets with sendograph metric

For a non-degenerate convex subset Y of the n  -dimensional Euclidean space Rⁿ${\mathbb{R}}^n$, let F(Y)$\mathbb{F}(Y)$ be the family of all fuzzy sets of Rⁿ${\mathbb{R}}^n$ which are upper semicontinuous, fuzzy convex and normal with compact supports contained in Y  . We show that the space F(Y)$\mathbb{F}(Y)$ with the topology of sendograph metric is homeomorphic to the separable Hilbert space ?²${?}^2$ if Y   is compact; and the space F(Rⁿ)$\mathbb{F}({\mathbb{R}}^n)$ is also homeomorphic to ?²${?}^2$.     

Compact imbeddings between variable exponent spaces with unbounded underlying domain

In this paper, we have established some compact imbedding theorems for some subspaces of W^1,p(x)(U)$W^{1,p\left(x\right)}\left(U\right)$ when the underlying domain U$U$ is unbounded. The domain we consider is mainly of type R^N(N≥2)$R^N(N\ge 2)$ or R^L×Ω(L≥2)$R^L\times \Omega (L\ge 2)$, where Ω⊂R^M$\Omega \subset R^M$ is a bounded domain with smooth boundary.     

Pullback attractors of non-autonomous stochastic degenerate parabolic equations on unbounded domains

This paper is concerned with pullback attractors of the stochastic p  -Laplace equation defined on the entire space Rⁿ${\mathbb{R}}^n$. We first establish the asymptotic compactness of the equation in L²(Rⁿ)$L^2({\mathbb{R}}^n)$ and then prove the existence and uniqueness of non-autonomous random attractors. This attractor is pathwise periodic if the non-autonomous deterministic forcing is time periodic. The difficulty of non-compactness of Sobolev embeddings on Rⁿ${\mathbb{R}}^n$ is overcome by the uniform smallness of solutions outside a bounded domain.     

The classification of locally conformally flat Yamabe solitons

This paper addresses the classification of locally conformally flat gradient Yamabe solitons. In the first part it is shown that locally conformally flat gradient Yamabe solitons with positive sectional curvature are rotationally symmetric. In the second part the classification of all radially symmetric gradient Yamabe solitons is given and their correspondence to smooth self-similar solutions of the fast diffusion equation on Rⁿ${\mathbb{R}}^n$ is shown. In the last section it is shown that any eternal solution to the Yamabe flow with positive Ricci curvature and with the scalar curvature attaining an interior space–time maximum must be a steady Yamabe soliton.