Radial solutions to the wave equation

We study some boundedness properties of radial solutions to the Cauchy problem associated to the wave equation (∂ _t ²-▵ _x )u(t,x)=0 and meanwhile we give a new proof of the solution formula. Received: July 7, 1998?Published online: March 19, 2002     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

Radial limits of positive solutions to the Darboux equation

Assume that a positive function u satisfies the Darboux equation

Rough hypoellipticity for the heat equation in Dirichlet spaces

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut-off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; $L^{\infty }$ structure results for ancient (local weak) solutions of the heat equation.     

Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems

Let B=B₁(0) be the unit ball in Rn and r=|x|. We study the poly-harmonic Dirichlet problem

     

Asymptotic behavior of solutions to the heat equation on noncompact symmetric spaces

This paper is twofold. The first part aims to study the long-time asymptotic behavior of solutions to the heat equation on Riemannian symmetric spaces $G/K$ of noncompact type and of general rank. We show that any solution to the heat equation with bi-K-invariant $L^1$ initial data behaves asymptotically as the mass times the fundamental solution, and provide a counterexample in the non bi-K-invariant case. These answer problems recently raised by J.L. Vázquez. In the second part, we investigate the long-time asymptotic behavior of solutions to the heat equation associated with the so-called distinguished Laplacian on $G/K$. Interestingly, we observe in this case phenomena which are similar to the Euclidean setting, namely $L^1$ asymptotic convergence with no bi-K-invariance condition and strong $L^{\infty }$ convergence.     

Stochastic porous media equation on general measure spaces with increasing Lipschitz nonlinearities

We prove the existence and uniqueness of probabilistically strong solutions to stochastic porous media equations driven by time-dependent multiplicative noise on a general measure space $(E,?\left(E\right),\mu )$, and the Laplacian replaced by a negative definite self-adjoint operator $L$. In the case of Lipschitz nonlinearities $\Psi$, we in particular generalize previous results for open $E?{\mathbb{R}}^d$ and $L=$Laplacian to fractional Laplacians. We also generalize known results on general measure spaces, where we succeeded in dropping the transience assumption on $L$, in extending the set of allowed initial data and in avoiding the restriction to superlinear behavior of $\Psi$ at infinity for $L^2\left(\mu \right)$-initial data.     

Commutators on Lipschitz spaces and related function spaces

We characterize the symbol functions so that the associated commutators with symbol functions and the Hilbert transform are bounded on Lipschitz space Λ _α ^p , where 1 < p < ∞ and 0 < α < 1/p. Properties of such symbols are also discussed.      

Convergence of difference solutions to generalized solutions of the Dirichlet problem for a second-order elliptic equation

Exact difference scheme operators are applied to construct a difference scheme for the Dirichlet problem for a secondorder elliptic equation with variable coefficients in a rectangle. If the solution belongs to the class W ₂ ² (), the scheme is of first-order accuracy in the grid norm of W ₂ ¹ ().Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 21–26, 1985.     

Solution of the Dirichlet problem for the Monge-Ampere equation in weight spaces

One proves the regular solvability of the problem: det(u_xx)=f(x,u,u_x)v>0 , k2, under the natural consistency conditions of the dimensions of the convex domain 0n and the growth of the function f(x,u,p) with respect to p.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 97–103, 1982.     

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any −1<m<0, positive functions f, g and u₀≥0, we prove that under some mild conditions on f, g and u₀ as R→∞ the solution u^R of the Dirichlet problem u_t=(u^m/m)_xx in (−R,R)×(0,∞), u(R,t)=(f(t)|m|R)^1/m, u(−R,t)=(g(t)|m|R)^1/m for all t>0, u(x,0)=u₀(x) in (−R,R), converges uniformly on every compact subset of R×(0,T) to the solution of the equation u_t=(u^m/m)_xx in R×(0,T), u(x,0)=u₀(x) in R, which satisfies some mass loss formula on (0,T) where T is the maximal time such that the solution u is positive. We also prove that the solution constructed is equal to the solution constructed in Hui (2007) [15] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.     

Littlewood-paley operators on the generalized Lipschitz spaces

Littlewood-Paley operators defined on a new kind of generalized Lipschitz spaces ₀ ^,p are studied. It is proved that the image of a function under the action of these operators is either equal to infinity almost everywhere or is in ₀ ^,p , where –n<<1 and 1<p<.     

解在QK 型及Dirichlet 型空间中的线性微分方程

对于单位圆盘上系数函数是解析函数的复微分方程
f⁽ⁿ⁾+A_n-1(z)f^(n-1)+…+A₁(z)f''+A₀(z)f=0,
给出了方程的系数函数和解函数之间的关系, 即当系数函数A_j 满足给定的条件时, 方程的所有解属于Q_K型空间和Dirichlet 型空间.     

Radial solutions with a vortex to an asymptotically linear elliptic equation

In this paper, we study a two-dimensional nonlinear elliptic equation:

Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains

For D, a bounded Lipschitz domain in Rⁿ, n ? 2, the classical layer potentials for Laplace's equation are shown to be invertible operators on L²(?D) and various subspaces of L²(?D). For 1 < p ? 2 and data in L^p(?D) with first derivatives in L^p(?D) it is shown that there exists a unique harmonic function, u, that solves the Dirichlet problem for the given data and such that the nontangential maximal function of ▽u is in L^p(?D). When n = 2 the question of the invertibility of the layer potentials on every L^p(?D), 1 < p < ∞, is answered.