Entire extensions and exponential decay for semilinear elliptic equations

We consider semilinear partial differential equations in ℝ ⁿ of the form

A spline collocation method for parabolic pseudodifferential equations

The purpose of this paper is to examine a boundary element collocation method for some parabolic pseudodifferential equations. The basic model problem for our investigation is the two-dimensional heat conduction problem with vanishing initial condition and a given Neumann or Dirichlet type boundary condition. Certain choices of the representation formula for the heat potential yield boundary integral equations of the first kind, namely the single layer and the hypersingular heat operator equations. Both of these operators, in particular, are covered by the class of parabolic pseudodifferential operators under consideration. Moreover, the spatial domain is allowed to have a general smooth boundary curve. As trial functions the tensor products of the smoothest spline functions of odd degree (space) and continuous piecewise linear splines (time) are used. Stability and convergence of the method is proved in some appropriate anisotropic Sobolev spaces.     

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form:

We analyze the singular convergence, as , in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:

(i)

We single out algebraic ``structure conditions' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.

(ii)

We deduce ``energy estimates ', uniformly in , by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions' on .

(iii)

We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

     

Local solvability for semilinear Fuchsian equations

We consider a semilinear elliptic operator P on a manifold B with a conical singular point. We assume P is Fuchs type in the linear part and has a non–linear lower order therms. Using the Schauder fixed point theorem, we prove the local solvability of P near the conical point in the weighted Sobolev spaces.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.     

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.     

Spectral triples and manifolds with boundary

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. We show that there is no tadpole for classical Dirac operators with a chiral boundary condition on spin manifolds.     

H-global well-posedness for semilinear wave equations

We consider the Cauchy problem for semilinear wave equations in with n?3. Making use of Bourgain's method in conjunction with the endpoint Strichartz estimates of Keel and Tao, we establish the H^s-global well-posedness with s<1 of the Cauchy problem for the semilinear wave equation. In doing so a number of nonlinear a priori estimates is established in the framework of Besov spaces. Our method can be easily applied to the case with n=3 to recover the result of Kenig-Ponce-Vega.     

Parabolic equations with measurable coefficients II

We prove the existence and uniqueness of solutions in Sobolev spaces to second-order parabolic equations in non-divergence form. The coefficients (except one of them) of second-order terms of the equations are measurable in both time and one spatial variables, and VMO (vanishing mean oscillation) in other spatial variables.     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Local energy decay for linear wave equations with variable coefficients

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L²-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].     

Partial differential equations with matricial coefficients and generalized translation operators

Let be the Bessel operator with matricial coefficients defined on by

where is a diagonal matrix and let be an matrix-valued function. In this work, we prove that there exists an isomorphism on the space of even , -valued functions which transmutes and . This allows us to define generalized translation operators and to develop harmonic analysis associated with . By use of the Riemann method, we provide an integral representation and we deduce more precise information on these operators.

     

Stochastic partial differential equations with unbounded coefficients and applications i

The present paper is the first instalment of a three-part study of stochastic partial differentia! equations (SPDEs) having unbounded coefficients. In this paper we prove existence and uniqueness theorems for a large class of parabolic SPDEs (having unbounded data), including a class of systems of SPDEs     

Regularity for quasi-linear degenerate elliptic equations with VMO coefficients

In this paper we establish an interior regularity of weak solution for quasi-linear degenerate elliptic equations under the subcritical growth if its coefficient matrix A（x, u） satisfies a VMO condition in the variable x uniformly with respect to all u, and the lower order item B（x, u, △↓u） satisfies the subcritical growth （1.2）. In particular, when F（x） ∈ L^q（Ω） and f（x） ∈ L^γ（Ω） with q,γ 〉 for any 1 〈 p 〈 ＋∞, we obtain interior HSlder continuity of any weak solution of （1.1） u with an index κ = min{1 - n/q, 1 - n/γ}.     

Degenerate principal series representations and their holomorphic extensions

Let X=H/L be an irreducible real bounded symmetric domain realized as a real form in an Hermitian symmetric domain D=G/K. The intersection S of the Shilov boundary of D with X defines a distinguished subset of the topological boundary of X and is invariant under H. It can be realized as S=H/P for certain parabolic subgroup P of H. We study the spherical representations of H induced from P. We find formulas for the spherical functions in terms of the Macdonald hypergeometric function. This generalizes the earlier result of Faraut-Koranyi for Hermitian symmetric spaces D. We consider a class of H-invariant integral intertwining operators from the representations on L²(S) to the holomorphic representations of G restricted to H. We construct a new class of complementary series for the groups H=SO(n,m), SU(n,m) (with n−m>2) and Sp(n,m) (with n−m>1). We realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of G=SU(n,m), SU(n,m)×SU(n,m) and SU(2n,2m) respectively.     

Higher order energy decay for damped wave equations with variable coefficients

Under appropriate assumptions the higher order energy decay rates for the damped wave equations with variable coefficients c(x)_utt−div(A(x)∇u)+a(x)_ut=0$c(x)u_{tt}-\mathrm{div}(A(x)\mathrm{\nabla }u)+a(x)u_t=0$ in Rⁿ${\mathbb{R}}^n$ are established. The results concern weighted (in time) and pointwise (in time) energy decay estimates. We also obtain weighted L²$L^2$ estimates for spatial derivatives.