Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

Acord transformatiu CRUE-CSICWe show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation −div(A∇u+bu)+c∇u+du=−divf+g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b,f∈L, [Formula presented] and c∈L for q≤∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b,d or c,d, we show a maximum principle and Moser's estimate for subsolutions with "good" constants. We also show the reverse Moser estimate for nonnegative supersolutions with "good" constants, under no smallness assumptions when q<∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation $-{\rm div}(A\nabla u+bu)+c\nabla u+du=-{\rm div}f+g$, assuming that $A$ is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions $b,f\in L^{n,1}$, $d,g\in L^{\frac{n}{2},1}$ and $c\in L^{n,q}$ for $q\leq\infty$, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with "good" constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on $b,d$ or $c,d$, we show a maximum principle and Moser's estimate for subsolutions with "good" constants. We also show the reverse Moser estimate for nonnegative supersolutions with "good" constants, under no smallness assumptions when $q<\infty$, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.Comment: 35 page

Minimizers for the thin one-phasefree boundary problem

We consider the "thin one-phase" free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in RnC1 C plus the area of the positivity set of that function in Rn . We establish full regularity of the free boundary for dimensions n (sic.) 2, prove almost everywhere regularity of the free boundary in arbitrary dimension, and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight. While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli in 1981. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments that are less reliant on the underlying PDE

Extensions of the Pesaran, Shin and Smith (2001) bounds testing procedure

We replicate the Pesaran et al. (J Appl Economet 16(1):289–326, 2001) bounds testing procedure (BTP) and extend it with 6 new cases, 4 of which involve a quadratic trend. We provide critical values for the BTP of the lagged regressors in levels under the framework of unrestricted error-correction models (UECMs) to account for degenerate cases of co-integration. Further, we extend the BTP with 11 cases for the quantile UECMs of Cho et al. (J Econom 188(1):281–300, 2015) and present critical values for interdecile and interquartile BTPs for the unrestricted cases. Moreover, we extend the Shin et al. (Festschrift in Honor of Peter Schmidt, Springer, New York, 2014) methodology to account for nonlinear, or asymmetric, responses of the dependent variables to its covariates (NARDL) and for distributional, or location, asymmetry (QARDL of Cho et al. 2015) of the dependent variable. This is the quantile nonlinear ARDL, or QNARDL. We provide codes that generate critical values for different sample sizes of the BTPs. These critical values are utilized in an empirical application of a dynamic equity valuation model for the S&P Global Index. Misspecifying a nonlinear relationship as linear produces misleading results and policy implications. There is strong evidence of (1) trading activity based on fundamentals and (2) the existence of a stable equilibrium relationship for the price-to-book (PB) ratio of the market index and its fundamentals. During periods of high PB relative to its fundamental values, convergence to equilibrium is faster than during periods of relatively low PB. There is also evidence of momentum trading, i.e., of traders that rely on positive feedback

Did the financial crisis affect the market valuation of large systemic U.S. banks?

We examine the impact of the financial crisis on the stock market valuation of large and systemic U.S. bank holding companies (BHCs). Using the Bertsatos and Sakellaris (2016) model of fundamental valuation of bank equity, we provide evidence that the financial crisis has not altered investors’ attitudes towards bank characteristics. In particular, before, during, and after the crisis, investors in large and systemic U.S. BHCs seemed to penalize leverage, albeit temporarily. Both before and after the crisis, they reward size in the short run. This pattern is appearing only briefly during the crisis. We also show that bank opacity plays no role in market valuation either in the short run or in the long run. Last but not least, we find evidence that stress testing has been informative to the market and that those BHCs that failed at the post-crisis stress tests were not subsequently valued differently by the market

Boundary Value Problems in Lipschitz Domains for Equations with Drifts

In this work we establish solvability and uniqueness for the $D_2$ Dirichlet problem and the $R_2$ Regularity problem for second order elliptic operators $L=-{\rm div}(A\nabla\cdot)+b\nabla\cdot$ in bounded Lipschitz domains, where $b$ is bounded, as well as their adjoint operators $L^t=-{\rm div}(A^t\nabla\cdot)-{\rm div}(b\,\cdot)$. The methods that we use are estimates on harmonic measure, and the method of layer potentials. The nature of our techniques applied to $D_2$ for $L$ and $R_2$ for $L^t$ leads us to impose a specific size condition on ${\rm div}b$ in order to obtain solvability. On the other hand, we show that $R_2$ for $L$ and $D_2$ for $L^t$ are uniquely solvable, assuming only that $A$ is Lipschitz continuous (and not necessarily symmetric) and $b$ is bounded