Sharp Regularity Research Articles

A sharp regularity result for the Euler equation with a free interface

A sharp regularity result for the Euler equation with a free interface

A Frobenius–Nirenberg theorem with parameter

Abstract The Newlander–Nirenberg theorem says that a formally integrable complex structure is locally equivalent to the complex structure in the complex Euclidean space. We will show two results about the Newlander–Nirenberg theorem with parameter. The first extends the Newlander–Nirenberg theorem to a parametric version, and its proof yields a sharp regularity result as Webster’s proof for the Newlander–Nirenberg theorem. The second concerns a version of Nirenberg’s complex Frobenius theorem and its proof yields a result with a mild loss of regularity.

Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Abstract We prove the W loc 2 ⁢ s , p ${W_{{\mathrm{loc}}}^{2s,p}}$ local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of ℝ N ${\mathbb{R}^{N}}$ . The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

Convolution Powers of Salem Measures With Applications

AbstractWe study the regularity of convolution powers for measures supported on Salemsets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for α of the form d/n, n = 2, 3, … there exist α-Salem measures for which the L2Fourier restriction theorem holds in the range. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular α-Salem measures, with sharp regularity results forn-fold convolutions for all n ∈ ℕ.

Examples of Improved Inversion of Different Airborne Electromagnetic Datasets Via Sharp Regularization

Large geophysical datasets are produced routinely during airborne surveys. The Spatially Constrained Inversion (SCI) is capable of inverting these datasets in an efficient and effective way by using a 1D forward modeling and, at the same time, enforcing smoothness constraints between the model parameters. The smoothness constraints act both vertically within each 1D model discretizing the investigated volume and laterally between the adjacent soundings. Even if the traditional, smooth SCI has been proven to be very successful in reconstructing complex structures, sometimes it generates results where the formation boundaries are blurred and poorly match the real, abrupt changes in the underlying geology. Recently, to overcome this problem, the original (smooth) SCI algorithm has been extended to include sharp boundary reconstruction capabilities based on the Minimum Support regularization. By means of minimization of the volume where, the spatial model variation is non-vanishing (i.e., the support of the variation), sharp-SCI promotes the reconstruction of blocky solutions. In this paper, we apply the novel sharp-SCI method to different types of airborne electromagnetic datasets and, by comparing the models against other geophysical and geological evidences, demonstrate the improved capabilities of in reconstructing sharp features.

Sharp regularity estimates for second order fully nonlinear parabolic equations

We prove sharp regularity estimates for viscosity solutions of fully nonlinear parabolic equations of the form Eq $$\begin{aligned} u_t - F\left( D^2u, Du, X, t\right) = f(X,t) \quad \text{ in } \quad Q_1, \end{aligned}$$ where F is elliptic with respect to the Hessian argument and $$f \in L^{p,q}(Q_1)$$ . The quantity $$\Xi (n, p, q) := \frac{n}{p}+\frac{2}{q}$$ determines to which regularity regime a solution of (Eq) belongs. We prove that when $$1< \Xi (n,p,q) < 2-\epsilon _F$$ , solutions are parabolically $$\alpha $$ -Holder continuous for a sharp, quantitative exponent $$0< \alpha (n,p,q) < 1$$ . Precisely at the critical borderline case, $$\Xi (n,p,q)= 1$$ , we obtain sharp parabolic Log-Lipschitz regularity estimates. When $$0< \Xi (n,p,q) <1$$ , solutions are locally of class $$C^{1+ \sigma , \frac{1+ \sigma }{2}}$$ and in the limiting case $$\Xi (n,p,q) = 0$$ , we show parabolic $$C^{1, \text {Log-Lip}}$$ regularity estimates provided F has “better” a priori estimates.

Sharp mean-square regularity results for SPDEs with fractional noise and optimal convergence rates for the numerical approximations

This article offers sharp spatial and temporal mean-square regularity results for a class of semi-linear parabolic stochastic partial differential equations (SPDEs) driven by infinite dimensional fractional Brownian motion with the Hurst parameter greater than one-half. In addition, mean-square numerical approximation of such problem are investigated, performed by the spectral Galerkin method in space and the linear implicit Euler method in time. The obtained sharp regularity properties of the problems enable us to identify optimal mean-square convergence rates of the full discrete scheme. These theoretical findings are accompanied by several numerical examples.

Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

AbstractWe study the positivity and regularity of solutions to the fractional porous medium equations in for m &gt; 1 and s ∈ (0,1), with Dirichlet boundary data u = 0 in and nonnegative initial condition .Our first result is a quantitative lower bound for solutions that holds for all positive times t &gt; 0. As a consequence, we find a global Harnack principle stating that for any t &gt; 0 solutions are comparable to ds/m, where d is the distance to ∂Ω. This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation.After this, we study the regularity of solutions. We prove that solutions are classical in the interior (C∞ in x and C1,α in t) and establish a sharp regularity estimate up to the boundary.Our methods are quite general and can be applied to wider classes of nonlocal parabolic equations of the form in Ω, both in bounded and unbounded domains.© 2016 Wiley Periodicals, Inc.

BMO solvability and the A∞ condition for second order parabolic operators

We prove that a sharp regularity property (A∞) of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. This equivalence was established in the elliptic case by Kenig, Kirchheim, Pipher and Toro, improving an earlier result of Kenig, Dindos and Pipher for solutions with data in BMO. The connection between regularity of the elliptic measure and certain Carleson measure properties of solutions was established in order to study solvability of boundary value problems for non-symmetric divergence form operators (Kenig, Koch, Pipher, and Toro). The extension to the parabolic setting requires an approach to the key estimates of the aforementioned works that primarily exploits the maximum principle. For various classes of parabolic operators ([24]), this criterion also provides an easier route to establish the solvability of the Dirichlet problem with data in Lp for some p, and also to quantify these results in several aspects.

A sharp regularization error estimate for bang‐bang solutions for an iterative Bregman regularization method for optimal control problems

AbstractIn the present work, we present numerical results for an iterative method for solving an optimal control problem with inequality contraints. The method is based on generalized Bregman distances. Under a combination of a source condition and a regularity condition on the active sets convergence results are presented. Furthermore we show by numerical examples that the provided a‐priori estimate is sharp in the bang‐bang case. (© 2016 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space

The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neumann type [13,15,26] with a series of new results in the case--so far rather unexplored--where the Neumann boundary term (input, control) possesses a regularity below $L^2(\Gamma)$ in space on the boundary $\Gamma$. We concentrate on the cases $H^{-\frac{1}{2}}(\Gamma))$, $H^{-\beta}(\Gamma))$, $H^{-1}(\Gamma))$, $\beta$ being a distinguished parameter of the problem. Our present results are consistent with the sharp result of [13,15,26] (obtained through a pseudo-differential/micro-local analysis approach), whose philosophy is expressed by a gain of $\beta$ in space regularity in going from the boundary control to the position in the interior. A number of physically relevant illustrations are given.

On the sharp regularity for arbitrary actions of nilpotent groups on the interval: the case of

In this work, we determine the largest $\unicode[STIX]{x1D6FC}$ for which the nilpotent group of four-by-four triangular matrices with integer coefficients and the value one in the diagonal embeds into the group of $C^{1+\unicode[STIX]{x1D6FC}}$ diffeomorphisms of the closed interval.

Sharp regularity and Cauchy problem of the spatially homogeneous Boltzmann equation with Debye–Yukawa potential

In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzmann equation with Debye–Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of distribution which contains the dual of the Sobolev spaces, there exists a unique solution which belongs to a more regular Sobolev space for any positive time. We also study the sharp regularity of the solution.

Analysis of Boundary Conditions for Crystal Defect Atomistic Simulations

Numerical simulations of crystal defects are necessarily restricted to finite computational domains, supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment. This work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed. We formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers. Using this foundation we then present rigorous error estimates for (i) a truncation method (Dirichlet boundary conditions), (ii) periodic boundary conditions, (iii) boundary conditions from linear elasticity, and (iv) boundary conditions from nonlinear elasticity. Numerical results confirm the sharpness of the analysis.

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

AbstractWe establish a frequency function monotonicity formula for two‐valued C1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C1, 1/2, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.

Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to Lu=f in Ω⊂Rn, L being the generator of any stable and symmetric Lévy process. Such nonlocal operators L depend on a finite measure on Sn−1, called the spectral measure.First, we study the interior regularity of solutions to Lu=f in B1. We prove that if f is Cα then u belong to Cα+2s whenever α+2s is not an integer. In case f∈L∞, we show that the solution u is C2s when s≠1/2, and C2s−ϵ for all ϵ>0 when s=1/2.Then, we study the boundary regularity of solutions to Lu=f in Ω, u=0 in Rn∖Ω, in C1,1 domains Ω. We show that solutions u satisfy u/ds∈Cs−ϵ(Ω‾) for all ϵ>0, where d is the distance to ∂Ω.Finally, we show that our results are sharp by constructing two counterexamples.

Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules. We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator. The power of the fractional exponent is exactly the same as the singular index of the non-cutoff collisional kernel of the Boltzmann equation. Therefore, we get the sharp regularity of solutions in the Gevrey class and also the sharp decay of solutions with an exponential weight. We also give a method to construct the solution of the Boltzmann equation by solving an infinite system of ordinary differential equations. The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.

Estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications

In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDEs around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform exterior ball condition. We prove rather sharp continuity results for these shape derivatives with respect to Sobolev norms of the boundary-traces of the displacements. With respect to previous results of this kind, the approach is quite different and is valid in any dimension N ≥ 2 . It is based on sharp regularity results for Poisson-type equations in such nonsmooth domains. We also enlarge the class of functionals and PDEs for which these estimates apply. Applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement.

Regularity of shadows and the geometry of the singular set associated to a Monge–Ampère equation

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally pro- jecting a convex set onto another. As an application we study the geometry and Hausdor dimension of the singular set corresponding to a Monge-Amp ere equation.

A Petrov--Galerkin Finite Element Method for Fractional Convection-Diffusion Equations

In this work, we develop variational formulations of Petrov--Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann--Liouville or Caputo derivative of order $\alpha\in(3/2, 2)$ in the leading term and both convection and potential terms. They arise in the mathematical modeling of asymmetric superdiffusion processes in heterogeneous media. The well-posedness of the formulations and sharp regularity pickup of the variational solutions are established. A novel finite element method (FEM) is developed, which employs continuous piecewise linear finite elements and “shifted” fractional powers for the trial and test space, respectively. The new approach has a number of distinct features: it allows the derivation of optimal error estimates in both the $L^2(D)$ and $H^1(D)$ norms; and on a uniform mesh, the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned. Further, in the Riemann--Liouville case, an enriched FEM is propose...