Weighted Sobolev Interpolation Inequalities on Certain Domains

We obtain some weighted Sobolev interpolation inequalities oncertain domains that include Lipschitz domains for doublingweights satisfying a weighted Poincaré inequality. Thesegeneralize most results in Gutierrez and Wheeden's paper [20].We also give some applications on Lipschitz domains for weightsof type dist (·, G)⁸, where G .     

Parabolic Schemes for Quasi-Linear Parabolic and Hyperbolic PDEs via Stochastic Calculus

We consider two quasi-linear initial-value Cauchy problems on ? ^d : a parabolic system and an hyperbolic one. They both have a first order non-linearity of the form φ(t, x, u)·?u, a forcing term h(t, x, u) and an initial condition u ₀ ∈ L ^∞(? ^d ) ∩ C ^∞(? ^d ), where φ (resp. h) is smooth and locally (resp. globally) Lipschitz in u uniformly in (t, x). We prove the existence of a unique global strong solution for the parabolic system. We show the existence of a unique local strong solution for the hyperbolic one and we give a lower bound regarding its blow up time. In both cases, we do not use weak solution theory but a direct construction based on parabolic schemes studied via a stochastic approach and a regularity result for sequences of parabolic operators. The result on the hyperbolic problem is performed by means of a non-classical vanishing viscosity method.     

A relative compactness criterion in Wiener–Sobolev spaces and application to semi-linear Stochastic PDEs

We prove a relative compactness criterion in Wiener–Sobolev space which represents a natural extension of the compact embedding of Sobolev space H¹ into , at the level of random fields. Then we give a specific statement of this criterion for random fields solutions of semi-linear stochastic partial differential equations with coefficients bounded in an appropriate way. Finally, we employ this result to construct solutions for semi-linear stochastic partial differential equations with distribution as final condition. We also give a probabilistic interpretation of this solution in terms of backward doubly stochastic differential equations formulated in a weak sense.     

Weighted Composition Operators from the Bloch Space to Weighted Banach Spaces on Bounded Homogeneous Domains

We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide ＂computable＂ estimates on the operator norm.     

Renormalized Solutions for Nonlinear Parabolic Systems with Three Unbounded Nonlinearities in Weighted Sobolev Spaces

We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form { b1（x,u1）/ t-div（a（x,t,u1,Du1））＋div（Ф1（u1））＋f1（x,u1,u2）=O in Q, b2（x,u2）/ t-div（a（x,t,u2,Du2））＋div（Ф2（u2））＋f2（x,u1,u2）=O in Q in the framework of weighted Sobolev spaces, where b（x,u） is unbounded function on u, the Carath6odory function ai satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function Фi is assumed to be continuous on ]R and not belong to （Lloc1（Q））N.     

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

Potential Analysis - We show that in a bounded simply connected planar domain Ω the smooth Sobolev functions Wk,∞(Ω) ∩ C∞(Ω) are dense in the homogeneous Sobolev...     

Parallel Solution of Stochastic PDEs

A stochastic partial differential equation is discretized by stochastic spectral finite elements. The resulting large linear system is solved by parallel iterative solvers, exploiting various forms of parallelism.     

Erratum to: An $$L_{p}$$-theory of Stochastic PDEs of Divergence Form on Lipschitz Domains

The purpose of this erratum is to correct the assumptions in Theorem 2.10 of [2] (Kim in J Theor Probab 22:220–238, 2009).     

A Note on Homogenization of Parabolic Equation in Perforated Domains

We are concerned with a class of parabolic equations in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes.By using the periodic unfolding method in perforated domains, we obtain the homogenization results under the conditions slightly weaker than those in the corresponding case considered by Nandakumaran and Rajesh(Nandakumaran A K, Rajesh M. Homogenization of a parabolic equation in perforated domain with Neumann boundary condition. Proc. Indian Acad. Sci.(Math. Sci.), 2002, 112(1): 195–207). Moreover,these results generalize those obtained by Donato and Nabil(Donato P, Nabil A. Homogenization and correctors for the heat equation in perforated domains. Ricerche di Matematica L. 2001, 50: 115–144).     

Boundary Behavior of Monotone Sobolev Functions in Orlicz Spaces on John Domains in a Metric Space

This paper deals with weighted boundary limits of monotone Sobolev functions in Orlicz spaces on bounded s-John domains in a metric space.     

Smooth Approximation of Sobolev Functions on Planar Domains

We examine two related problems concerning a planar domain .The first is whether Sobolev functions on can be approximatedby global C functions, and the second is whether approximationcan be done by functions in C() which, together with all derivatives,are bounded on . We find necessary and sufficient conditionsfor certain types of domains, such as starshaped domains, andwe construct several examples which show that the general problemis quite difficult, even in the simply connected case.     

加权p-Laplace方程解的整体分叉问题

该文利用指标理论，谱理论以及标准分叉定理，研究了加权索伯列夫空间中的p Laplace方程初值问题的解的整体分叉现象。     

Viscosity Solutions to Second Order Parabolic PDEs on Riemannian Manifolds

In this work we consider viscosity solutions to second order parabolic PDEs u _t +F(t,x,u,du,d ² u)=0 defined on complete Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold M has nonnegative sectional curvature, we get the finest results. If one additionally requires F to depend on d ² u in a uniformly continuous manner, the assumptions on curvature can be thrown away.     

A Remark on the Sobolev Regularity of Classical Solutions to Uniformly Parabolic Equations

We prove that C^2+α,1+α/2 (Q?) solutions of problem (1.6) below are in a subspace H_c^m+2(Q) of H^{m+2,(m+2)/2(Q)} for all m ∈ ?, if f and the coefficients are in H_c^m(Q)∪C^α,α/2 (Q?). We apply this result to obtain global existence of Sobolev solutions to the quasilinear problem (1.26) below.     

一个抛物型方程侧边值问题的正则逼近解在一类Sobolev空间中的最优误差界

逆热传导问题(IHCP)是严重不适定问题,即问题的解(如果存在)不连续依赖于数据．但目前关于逆热传导问题的已有结果主要是针对标准逆热传导问题．文中给出了出现在实际问题中的一个抛物型方程侧边值问题,即一个含有对流项的非标准型逆热传导问题的正则逼近解一类Sobolev空间中的最优误差界．     

A Stochastic Subgradient Method for Distributionally Robust Non-convex and Non-smooth Learning

Journal of Optimization Theory and Applications - We consider a distributionally robust formulation of stochastic optimization problems arising in statistical learning, where robustness is with...     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

关于变指数加权Sobolev空间的拟连续性

本文给出了一些关于变指数加权Sobolev空间拟连续性的精确刻画. 进而在拟连续的意义下得到变指数加权Sobolev空间唯一性结果.     

有界对称域上的加权Plancherel公式

本文给出加权 Ｐｌａｎｃｈｅｒｅｌ公式与Ｈｅｒｍｉｔｅ对称空间上的齐性线从上Ｐｌａｎｃｈｅｒｅｌ公式的关系，由此导出一般有界对称域上的加权Ｐｌａｎｃｈｅｒｅｌ公式．