10042269
On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains
We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobole spaces $\mathfrak{H}^{\gamma,q}_{p,\theta } (\mathcal{O},T)$ . The summability parameters $p$ and $q$ in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H o lder regularity in time is analysed. Moreover, we prove a general embedding of weighte $L_p(\mathcal{O})$ - Sobolev spaces into the scale o Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$ , $1/\tau=\alpha/d+1/p$ , $\alpha>0$ . This leads to a H o lder- Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.
121084268
A Sobolev space theory of SPDEs with constant coefficients on a half line
Equations of the form $du=(a^{ij}u_{x^{i}x^{j}} +D_{i}f^{i})\,dt+\sum_{k}(\sigma^{ik}u_{x^{i}} +g^{k})\,dw^{k}_{t}$ are considered for t > 0 and $x\in\bR^{d}_{+}$. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power $p\in[2,\infty)$.
16838339
A Sobolev Space Theory of SPDEs with Constant Coefficients in a Half Space
Equations of the form du = (auxixj +Dif i) dt+ ∑ k (σuxi + g k) dwk t are considered for t > 0 and x ∈ R+. The unique solvability of these equations is proved in weighted Sobolev spaces with fractional positive or negative derivatives, summable to the power p ∈ [2,∞).
209866254
ON STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS IN C DOMAINS
Stochastic partial differential equations with variable coefficients are considered in C domains. Existence and uniqueness results are given in Sobolev spaces with weights allowing the derivatives of the solutions to blow up near the boundary. The number of derivatives of the solution can be negative and fractional, and the coefficients of the equations are allowed to substantially oscillate or blow up near the boundary.
119311074
A Weighted Sobolev Space Theory of Parabolic Stochastic PDEs on Non-smooth Domains
In this article, we study parabolic stochastic partial differential equations (see Eq. (1.1)) defined on arbitrary bounded domain $$\mathcal{O }\subset \mathbb{R }^d$$ admitting the Hardy inequality 0.1$$\begin{aligned} \int _{\mathcal{O }}|\rho ^{-1}g|^2\,\text{ d}x\le C\int _{\mathcal{O }}|g_x|^2 \text{ d}x, \quad \forall g\in C^{\infty }_0(\mathcal{O }), \end{aligned}$$where $$\rho (x)=\text{ dist}(x,\partial \mathcal{O }).$$ Existence and uniqueness results are given in weighted Sobolev spaces $$\mathfrak{H }_{p,\theta }^{\gamma }(\mathcal{O },T),$$ where $$p\in [2,\infty ), \gamma \in \mathbb{R }$$ is the number of derivatives of solutions and $$\theta $$ controls the boundary behavior of solutions (see Definition 2.5). Furthermore, several Hölder estimates of the solutions are also obtained. It is allowed that the coefficients of the equations blow up near the boundary.
56085126
Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.
55717678
Adaptive wavelet methods for the stochastic Poisson equation
We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝd. The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.
122324791
Sobolev space theory of SPDEs with continuous or measurable leading coefficients
We study stochastic partial differential equations with variable coefficients defined on and bounded C1 domains. For equations with continuous leading coefficients we give existence and uniqueness results in Lq(Lp)-spaces, where it is allowed for the powers of summability with respect to space and time variables to be different. For equations with measurable leading coefficients we give unique solvability in Lp-spaces.
197483765
Maximal $L^p$-regularity for stochastic evolution equations
We prove maximal $L^p$-regularity for the stochastic evolution equation 
\[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], 
U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12\pi$ on a space $L^q(\mathcal{O},\mu)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space $\XAp $ we prove existence of unique strong solution with trajectories in \[L^p(0,T;\Dom(A))\cap C([0,T];\XAp),\] provided the non-linearities $F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},\mu)$ and $B:[0,T]\times \Dom(A) \to \g(H,\Dom(A^{\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. 
Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain $\OO\subseteq \R^d$ with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\OO))^d$ is shown.
73715217
SOBOLEV SPACES WITH WEIGHTS IN DOMAINS AND BOUNDARY VALUE PROBLEMS FOR DEGENERATE ELLIPTIC EQUATIONS
A family of Banach spaces is introduced to control the interior smoothness and boundary behavior of functions in a general domain. Interpolation, embedding, and other properties of the spaces are studied. As an application, a certain degenerate second-order elliptic partial differential equation is considered.
121409820
Lp solvability of divergence type parabolic and elliptic systems with partially BMO coefficients
We prove the $${{\mathcal{H}}^{1}_{p,q}}$$ solvability of second order systems in divergence form with leading coefficients Aαβ only measurable in (t, x1) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A11 is measurable in t and has a small BMO semi-norm in the other variables; (ii) A11 is measurable in x1 and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition.
14910493
Stochastic integration in quasi-Banach spaces
In this paper we develop a stochastic integration theory for processes with values in a quasi-Banach space. The integrator is a cylindrical Brownian motion. The main results give sufficient conditions for stochastic integrability. They are natural extensions of known results in the Banach space setting. We apply our main results to the stochastic heat equation where the forcing terms are assumed to have Besov regularity in the space variable with integrability exponent $p\in (0,1]$. The latter is natural to consider for its potential application to adaptive wavelet methods for stochastic partial differential equations.
121377515
Besov regularity for elliptic boundary value problems
This paper studies the regularity of solutions to boundary value problems for the Laplace operator on Lipschitz domains {Omega} in R{sup d} and its relationship with adaptive and other nonlinear methods for approximating these solutions. The smoothness spaces which determine the efficiency of such nonlinear approximation in L{sub p}({Omega}) are the Besov spaces B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})), {tau} := ({alpha}/d + 1/p){sup -1}. Thus, the regularity of the solution in T this scale of Besov spaces is investigated with the aim of determining the largest a for which the solution is in B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})). The regularity theorems given in this paper build upon the recent results of Jerison and Kenig. The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions. 14 refs., 1 fig.
123259721
Besov regularity for second order elliptic boundary value problems with variable coefficients
This paper is concerned with some theoretical foundations for adaptive numerical methods for elliptic boundary value problems. The approximation order that can be achieved by such an adaptive method is determined by certain Besov regularity of the weak solution. We study Besov regularity for second order elliptic problems in bounded domains in ℝd. The investigations are based on intermediate Schauder estimates and on some potential theoretic framework. Moreover, we use characterizations of Besov spaces by wavelet expansions.
121140677
Besov regularity for elliptic boundary value problems in polygonal domains
Abstract This paper is concerned with the regularity of the solutions to elliptic boundary value problems in polygonal domains Ω contained in R 2. Especially, we consider the specific scale B τ α (L τ (Ω)) , 1 τ = α 2 + 1 p , of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. The proofs are based on specific representations of the solutions which were, e.g., derived by Grisvard [1], and on characterizations of Besov spaces by wavelet expansions.
20155765
Besov regularity of edge singularities for the Poisson equation in polyhedral domains
This paper is concerned with the regularity of the solutions to the Poisson equation in polyhedral domains Ω contained in R3. Especially, we consider the specific scale Bτα(Lτ(Ω)),1/τ = α/3 + ½, of Besov spaces. The regularity of the variational solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and non-linear numerical schemes. It is well-known that in polyhedral domains different types of singularities according to edges and vertices occur. In this paper, we shall primarily be concerned with the Besov regularity of edge singularities. By the usual reduction procedure, these singularity functions are studied in an unfinite cylinder R × ϒ, where ϒ denotes a bounded polygonal domain contained in R2. We show that the singularity functions are much smoother in the specific Besov scale than in the usual L2–Sobolov scale which justifies the use of adaptive schemes. The proofs are based on specific representations of the solutions and on characterizations of Besov spaces by wavelet expansions. Copyright © 2002 John Wiley & Sons, Ltd.
37578445
Besov Regularity for the Poisson Equation in Smooth and Polyhedral Cones
The regularity of solutions to the Dirichlet and Neumann problems in smooth and polyhedral cones contained in R 3 is studied with particular attention to the specific scale B s T(L T), 1/t = s/3 + 1/2, of Besov spaces. The regularity of the solution in these Besov spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We show that the solutions are much smoother in the specific Besov scale than in the usual L 2-Sobolev scale, which justifies the use of adaptive schemes. The proofs are performed by combining weighted Sobolev estimates with characterizations of Besov spaces by wavelet expansions.
123628431
Parabolic Besov Regularity for the Heat Equation
We obtain parabolic Besov smoothness improvement for temperatures on cylindrical regions based on Lipschitz domains. The results extend those for harmonic functions obtained by S. Dahlke and R. DeVore using the wavelet description of Besov regularity.
119660410
An Lp-Theory of SPDEs on Lipschitz Domains
Stochastic partial differential equations are considered on Lipschitz domains. Existence and uniqueness results are given in weighted Sobolev spaces, and Hölder estimates of the solutions are also obtained. The number of derivatives of the solutions can be any real number, in particular, it can be negative and fractional. It is allowed that the coefficients of the equations blow up near the boundary.
118840232
Semigroups of Linear Operators and Applications to Partial Differential Equations
1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.
13833114
Maximal Lp-regularity for the Laplacian on Lipschitz domains
We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following two domains of definition: $${{D_1(\Delta)=\{u\in W^{1,p}(\Omega): \Delta u\in L^p(\Omega),\ Bu=0\}}}$$ , or $${{D_2(\Delta)=\{u\in W^{2,p}(\Omega)}: {Bu=0\}}}$$ , where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on Lp(Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and $${{1 < p \leq 2}}$$ , the Laplacian with domain D2(Δ) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that proves that, in general, the Dirichlet–Laplacian with domain D1(Δ) is not even a closed operator.
48351350
Biorthogonal bases of compactly supported wavelets
Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient condition for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.
2489923
On Restrictions and Extensions of the Besov and Triebel–Lizorkin Spaces with Respect to Lipschitz Domains
The restrictions B s pq (Ω) and F s pq (Ω) of the Besov and Triebel–Lizorkin spaces of tempered distributions B s pq (ℝ n ) and F s pq (ℝ n ) to Lipschitz domains Ω⊂ℝ n are studied. For general values of parameters ( s ∈ℝ, p >0, q >0) a ‘universal’ linear bounded extension operator from B s pq (Ω) and F s pq (Ω) into the corresponding spaces on ℝ n is constructed. The construction is based on a new variant of the Calderon reproducing formula with kernels supported in a fixed cone. Explicit characterizations of the elements of B s pq (Ω) and F s pq (Ω) in terms of their values in Ω are also obtained.
123880057
Intrinsic characterizations of Besov spaces on Lipschitz domains
The aim of this paper is to study the equivalence between quasi-norms of Besov spaces on domains. We suppose that the domain Ω ⊂ ℝn is a bounded Lipschitz open subset in ℝn. First, we define Besov spaces on Ω as the restrictions of the corresponding Besov spaces on ℝn. Then, with the help of equivalent and intrinsic characterizations (the Peetre-type characterization 3.10 and the characterization via local means 3.13) of these spaces, we get another equivalent and intrinsic quasi-norm using, this time, generalized differences and moduli of smoothness. We extend the well-known characterization of Besov spaces on ℝn described in Theorem 2.4 to the case of Lipschitz domains.
121487101
Maximal functions measuring smoothness
Maximal functions which measure the smoothness of a function are intro" duced and studied from the point of view of their relationship to classical smoothness and their use in proving embedding theorems, extension theorems and various results on differentiation. New spaces of functions which generalize Sobolev spaces are introduced. AMS(MOS) subject classifications (1980). Primary 26B35, 46E35, 26Al5 Secondary 42B25