Resolvent Estimates in L^p for the Stokes Operator in Lipschitz Domains

We establish the $L^p$ resolvent estimates for the Stokes operator in Lipschitz domains in $R^d$, $d\ge 3$ for $|\frac{1}{p}-1/2|< \frac{1}{2d} +\epsilon$. The result, in particular, implies that the Stokes operator in a three-dimensional Lipschitz domain generates a bounded analytic semigroup in $L^p$ for (3/2)-\varep < p< 3+\epsilon. This gives an affirmative answer to a conjecture of M. Taylor.Comment: 28 page. Minor revision was made regarding the definition of the Stokes operator in Lipschitz domain

Harnack inequality and regularity for degenerate quasilinear elliptic equations

We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,\alpha}$ local estimates for solutions of a degenerate equation in non divergence form

Green's functions for parabolic systems of second order in time-varying domains

We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and 1/2-H\"older continuous in the time variable, under the assumption that weak solutions of the system satisfy an interior H\"older continuity estimate. We also derive global pointwise estimates for Green's function in such time-varying domains under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate and a local H\"older continuity estimate. In particular, our results apply to complex perturbations of a single real equation.Comment: 25 pages, 0 figur

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II

We continue the development, by reduction to a first order system for the conormal gradient, of $L^2$ \textit{a priori} estimates and solvability for boundary value problems of Dirichlet, regularity, Neumann type for divergence form second order, complex, elliptic systems. We work here on the unit ball and more generally its bi-Lipschitz images, assuming a Carleson condition as introduced by Dahlberg which measures the discrepancy of the coefficients to their boundary trace near the boundary. We sharpen our estimates by proving a general result concerning \textit{a priori} almost everywhere non-tangential convergence at the boundary. Also, compactness of the boundary yields more solvability results using Fredholm theory. Comparison between classes of solutions and uniqueness issues are discussed. As a consequence, we are able to solve a long standing regularity problem for real equations, which may not be true on the upper half-space, justifying \textit{a posteriori} a separate work on bounded domains.Comment: 76 pages, new abstract and few typos corrected. The second author has changed nam

Localization on a quantum graph with a random potential on the edges

We prove spectral and dynamical localization on a cubic-lattice quantum graph with a random potential. We use multiscale analysis and show how to obtain the necessary estimates in analogy to the well-studied case of random Schroedinger operators.Comment: LaTeX2e, 18 page

Maximal L p -regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains ?, both with the following two domains of definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) : 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(?) which implies maximal regularity for the corresponding Cauchy problems. In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain D2(?) 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(?) is not even a closed operator

The role of parental achievement goals in predicting autonomy-supportive and controlling parenting

Although autonomy-supportive and controlling parenting are linked to numerous positive and negative child outcomes respectively, fewer studies have focused on their determinants. Drawing on achievement goal theory and self-determination theory, we propose that parental achievement goals (i.e., achievement goals that parents have for their children) can be mastery, performance-approach or performance-avoidance oriented and that types of goals predict mothers' tendency to adopt autonomy-supportive and controlling behaviors. A total of 67 mothers (aged 30-53 years) reported their goals for their adolescent (aged 13-16 years; 19.4 % girls), while their adolescent evaluated their mothers' behaviors. Hierarchical regression analyses showed that parental performance-approach goals predict more controlling parenting and prevent acknowledgement of feelings, one autonomy-supportive behavior. In addition, mothers who have mastery goals and who endorse performance-avoidance goals are less likely to use guilt-inducing criticisms. These findings were observed while controlling for the effect of maternal anxiety

Regularity of Infinity for Elliptic Equations with Measurable Coefficients and Its Consequences

This paper introduces a notion of regularity (or irregularity) of the point at infinity for the unbounded open subset of \rr^{N} concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the A-harmonic measure of the point at infinity is zero (or positive). A necessary and sufficient condition for the existence of a unique bounded solution to the Dirichlet problem in an arbitrary open set of \rr^{N}, N\ge 3 is established in terms of the Wiener test for the regularity of the point at infinity. It coincides with the Wiener test for the regularity of the point at infinity in the case of Laplace equation. From the topological point of view, the Wiener test at infinity presents thinness criteria of sets near infinity in fine topology. Precisely, the open set is a deleted neigborhood of the point at infinity in fine topology if and only if infinity is irregular.Comment: 20 page

Children’s coping with in vivo peer rejection: An experimental investigation

We examined children's behavioral coping in response to an in vivo peer rejection manipulation. Participants (N=186) ranging between 10 and 13 years of age, played a computer game based on the television show Survivor and were randomized to either peer rejection (i.e., being voted out of the game) or non-rejection control. During a five-min. post-feedback waiting period children's use of several behavioral coping strategies was assessed. Rejection elicited a marked shift toward more negative affect, but higher levels of perceived social competence attenuated the negative mood shift. Children higher in depressive symptoms were more likely to engage in passive and avoidant coping behavior. Types of coping were largely unaffected by gender and perceived social competence. Implications are discussed. © 2006 Springer Science+Business Media, LLC