Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

On a parabolic problem with nonlinear term in a half space and global behavior of solutions

We take up the existence and global behavior of positive continuous solutions of the following nonlinear parabolic equation in (n?2) with boundary conditions u=0 on and u(x,0)=u₀(x). The nonlinear term is required to satisfy some conditions related to a functional class , which we introduce in this paper and will be called parabolic Kato class in the half space. Our approach is based on potential theory.     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Sublinear singular elliptic problems with two parameters

We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on . The proofs are based on various techniques related to the maximum principle for elliptic equations.     

The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

By constructing the comparison functions and the perturbed method, it is showed that any solution u∈C²(Ω) to the semilinear elliptic problems Δu=k(x)g(u), x∈Ω, u_|∂Ω=+∞ satisfies , where Ω is a bounded domain with smooth boundary in RN; , −2<σ, c₀>0, ; g∈C¹[0,∞), g?0 and is increasing on (0,∞), there exists ρ>0 such that , ∀ξ>0, , .     

Convergence of discrete schemes for the Perona-Malik equation

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation ut=(?^′x(ux)), , when the initial datum is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when has a whole interval where is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing ?(⋅) with the truncated function min(?(⋅),1), and it turns out that u solves a free boundary problem. The free boundary consists of the points dividing the region where |ux|>1 from the region where |ux|?1. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals , i.e., the standing solution of the convexified problem.     

Global existence and asymptotic behavior of solutions for nonlinear parabolic equations on unbounded domains

We study the existence and the asymptotic behavior of positive solutions for the parabolic equation on D×(0,∞), where is a some unbounded domain in and V belongs to a new parabolic class J^∞ of singular potentials generalizing the well-known parabolic Kato class at infinity P^∞ introduced recently by Zhang. We also show that the choice of this class is essentially optimal.     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Positivity properties in noncommutative convolution algebras with applications in pseudo-differential calculus

We study , of all such that for every ?∈C^∞₀, where denotes the twisted convolution. We prove that certain boundedness for are completely determined of the behaviour for a at origin, for example that , and that if a(0)<∞, then a∈L²∩L^∞. We use the results in order to determine wether positive pseudo-differential operators belong to certain Schatten-casses or not.     

On controlled extensions of functions

For any numerical function we give sufficient conditions for resolving the controlled extension problem for a closed subset A of a normal space X. Namely, if the functions , and satisfy the equality E(f(a),g(a))=h(a), for every a∈A, then we are interested to find the extensions f? and ? of f and g, respectively, such that , for every x∈X. We generalize earlier results concerning E(u,v)=u·v by using the techniques of selections of paraconvex-valued LSC mappings and soft single-valued mappings.     

A further three critical points theorem

If X is a real Banach space, we denote by W_X the class of all functionals possessing the following property: if {u_n} is a sequence in X converging weakly to u∈X and lim inf_n→∞Φ(u_n)≤Φ(u), then {u_n} has a subsequence converging strongly to u.In this paper, we prove the following result:Let X be a separable and reflexive real Banach space; an interval; a sequentially weakly lower semicontinuous C¹ functional, belonging to W_X, bounded on each bounded subset of X and whose derivative admits a continuous inverse on X^∗; a C¹ functional with compact derivative. Assume that, for each λ∈I, the functional Φ−λJ is coercive and has a strict local, not global minimum, say .Then, for each compact interval [a,b]⊆I for which , there exists r>0 with the following property: for every λ∈[a,b] and every C¹ functional with compact derivative, there exists δ>0 such that, for each μ∈[0,δ], the equation

Φ^′(x)=λJ^′(x)+μΨ^′(x)     

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

Let Γ be a d-bounded distance-regular graph with diameter d?3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L^′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L^′(x,i) by inclusion or reverse inclusion, L^′(x,i) is denoted by or . We prove that and are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of