Elliptic Estimates Research Articles

The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant

We study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 𝕋3. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state [Formula: see text], [Formula: see text]. The dust-Einstein system is the case cs = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases [Formula: see text], which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions.

Global Existence of Smooth Solutions and Convergence to Barenblatt Solutions for the Physical Vacuum Free Boundary Problem of Compressible Euler Equations with Damping

AbstractFor the physical vacuum free boundary problem with the sound speed being C1/2‐Hölder continuous near vacuum boundaries of the one‐dimensional compressible Euler equations with damping, the global existence of the smooth solution is proved, which is shown to converge to the Barenblatt self‐similar solution for the porous media equation with the same total mass when the initial datum is a small perturbation of the Barenblatt solution. The pointwise convergence with a rate of density, the convergence rate of velocity in the supremum norm, and the precise expanding rate of the physical vacuum boundaries are also given. The proof is based on a construction of higher‐order weighted functionals with both space and time weights capturing the behavior of solutions both near vacuum states and in large time, an introduction of a new ansatz, higher‐order nonlinear energy estimates, and elliptic estimates.© 2016 Wiley Periodicals, Inc.

Three-point galaxy-galaxy lensing as a probe of dark matter halo shapes

We propose a method to measure the ellipticities of dark matter halos using the lens-shear-shear 3-point correlation function. This method is immune to effects of galaxy-halo misalignments that can potentially limit 2-point galaxy-galaxy lensing measurements of halo anisotropy. Using a simple model for the projected mass distributions of dark matter halos, we construct an ellipticity estimator that sums over all possible triangular configurations of the 3-point function. By applying our estimator to halos from N-body simulations, we find that systematic errors in the recovered ellipticity will be at the ≲ 5% fractional level. We estimate that future imaging surveys like LSST will have sufficient statistics to detect halo ellipticities using 3-point lensing.

On annealed elliptic Green's function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb Z^d$. The distribution $\langle \cdot \rangle $ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green's function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ and $L^1$, respectively, in probability, i.e., they obtained bounds on $\smash {\langle |\nabla _x G(t,x,y)|^2\rangle ^{{1}/{2}}}$ and $\langle |\nabla _x \nabla _y G(t,x,y)|\rangle $. In particular, the elliptic Green's function $G(x,y)$ satisfies optimal annealed bounds. In their recent work, the authors extended these elliptic bounds to higher moments, i.e., $L^p$ in probability for all $p<\infty $. In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates for $\langle |\nabla _x G(x,y)|^2\rangle ^{{1}/{2}}$ and $\langle |\nabla _x \nabla _y G(x,y)|\rangle $.

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.

Existence of the solution to electromagnetic wave scattering problem for an impedance body of an arbitrary shape

A new proof is given for the existence of the solution to electromagnetic (EM) wave scattering problem for an impedance body of an arbitrary shape. The proof is based on the elliptic systems theory and elliptic estimates for the solutions of such systems.

Existence and non existence results for supercritical systems of Liouville-type equations on simply connected domains

We obtain a Pohozaev-type identity which yields a generalization to the systems case of the well known scalar non-existence threshold for Liouville-type mean field equations on strictly starshaped domains. These newly derived non-existence results suggest that in principle solutions could be find in a region of parameters far away from the subcritical regime with respect to the vectorial Moser–Trudinger and Log-HLS inequalities found by Chipot, Shafrir and Wolansky. Indeed, we succeed in proving that the Dirichlet problem for cooperative Liouville systems admits solutions on “thin” simply connected domains in the supercritical regime. This is an improvement of the existence theory for cooperative Liouville systems since in that region solutions were known to exist only on multiply connected domains. Finally, combining spectral elliptic estimates and Orlicz-spaces techniques with a trick introduced by Wolansky we prove that these newly derived solutions are strict local minimizers of the Moser–Trudinger-type and free-energy functionals.

A product property of Sobolev spaces with application to elliptic estimates

In this paper a Sobolev inequality, which generalizes the ordinary Banach algebra property of such spaces, is established; for p\in[1,\infty) , n,m\in\mathbb Z^+ , and m\ge2 that satisfy m&gt;n/p , \left\Vert\phi\psi\right\Vert_{m,p,\Omega}\le K \left[ \left( \sup_{\Omega_s} \left\vert \phi \right\vert \right) \left\Vert\psi\right\Vert_{m,p,\Omega} + \Bigl(\left\Vert\psi\right\Vert_{m-1,q,\Omega}+\left\Vert\psi\right\Vert_{m-1,p,\Omega}\Bigr)\right] \left\Vert\phi\right\Vert_{m,p,\Omega} for all \phi,\psi \in W^{m,p}(\Omega) that satisfy \rm {spt} \psi \subset \Omega_s\subset \Omega and domains \Omega\subset\mathbb R^n that are nonempty, open, and satisfy the cone condition. Here q=p if p&gt;n , q\in (n/\Upsilon,pn/(n-p)] if n&gt;p , q\in (n/\Upsilon,\infty) if p=n , K=K(n,p,m,q,\mathcal C) , where \mathcal C is the cone from the cone condition, and \Upsilon:= [[{n/p}]] , the largest integer less than or equal to n/p .

Elliptic estimates in composite media with smooth inclusions: an integral equation approach

We consider a scalar elliptic equation for a composite medium consisting of homogeneous C^{1, α0} inclusions, 0< α0≤ 1, embedded in a constant matrix phase. When the inclusions are separated and are separated from the boundary, the solution has an integral representation, in terms of potential functions defined on the boundary of each inclusion. We study the system of integral equations satisfied by these potential functions as the distance between two inclusions tends to 0. We show that the potential functionsunctions converge in C^{0,α}, 0< α < α0 to limiting potential functions, with which one can represent the solution when the inclusions are touching. As a consequence, we obtain uniform C^{1,α} bounds on the solution, which are independent of the inter-inclusion distances.

Capillary–Gravity Water Waves with Discontinuous Vorticity: Existence and Regularity Results

In this paper we construct periodic capillarity-gravity water waves with an arbitrary bounded vorticity distribution. This is achieved by reexpressing, in the height function formulation of the water wave problem, the boundary condition obtained from Bernoulli's principle as a nonlocal differential equation. This enables us to establish the existence of weak solutions of the problem by using elliptic estimates and bifurcation theory. Secondly, we investigate the a priori regularity of these weak solutions and prove that they are in fact strong solutions of the problem, describing waves with a real-analytic free surface. Moreover, assuming merely integrability of the vorticity function, we show that any weak solution corresponds to flows having real-analytic streamlines.

Elliptic interpolation estimates for non-standard growth operators

We derive a class of potential estimates for elliptic equations with non-standard growth having a measure on the right-hand side; in particular our results allow to interpolate between the pointwise estimates available for solutions to this equation and the ones for the gradient. We allow bounded coefficients in BMO and VMO classes.

The Viscous Surface-Internal Wave Problem: Global Well-Posedness and Decay

We consider the free boundary problem for two layers of immiscible, viscous, incompressible fluid in a uniform gravitational field, lying above a general rigid bottom in a three-dimensional horizontally periodic setting. We establish the global well-posedness of the problem both with and without surface tension. We prove that without surface tension the solution decays to the equilibrium state at an almost exponential rate; with surface tension, we show that the solution decays at an exponential rate. Our results include the case in which a heavier fluid lies above a lighter one, provided that the surface tension at the free internal interface is above a critical value, which we identify. This means that sufficiently large surface tension stabilizes the Rayleigh–Taylor instability in the nonlinear setting. As a part of our analysis, we establish elliptic estimates for the two-phase stationary Stokes problem.

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semi-linear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori L ∞-bounds and sub- and super-solutions to apply a fixed point argument. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, “approximating” problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed non-linear problems with irregular data.

Regularity for quasilinear equations on degenerate singular sets

We prove a new universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at singular set of degeneracy $$\fancyscript{S}(u) := \{X : D u(X) = 0 \}$$ . Our main Theorem reveals that along $$\fancyscript{S}(u)$$ , $$u$$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $$\fancyscript{S}(u)$$ , $$Du$$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that $$H^1$$ -weak solutions to $$\text {div}\left( a_{ij}(X)Du \right) = 0$$ , with $$a_{ij}$$ elliptic and Dini-continuous are actually $$C^{1,1^{-}}$$ along $$\fancyscript{S}(u)$$ . The results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

Accuracy and Precision of Selected Wound Area Measurement Methods in Diabetic Foot Ulceration

The present study assesses precision and accuracy of four selected methods of wound area measurements in diabetic foot wounds. The areas of 16 wound shapes were measured using linear measurement with elliptical estimation (Elliptical method), using the Visitrak™ (Smith & Nephew, London, United Kingdom) device, the SilhouetteMobile™ device (ARANZ Medical Ltd., Christchurch, New Zealand), and the TeleDiaFoS system (Nalecz Institute of Biocybernetics and Biomedical Engineering, Warsaw, Poland). The actual area of a wound shape was determined with a tested scanner and specifically developed software. Accuracy of the area measurement method was assessed by the relative error (RE), whereas precision was assessed by the coefficient of variation (CV). The overall absolute REs were 13.3%, 6.8%, 2.1%, and 2.3% for the Elliptical method, the Visitrak device, the TeleDiaFoS system, and the SilhouetteMobile device, respectively. The accuracy of the Visitrak device was remarkably reduced for wound areas smaller than 2 cm². The overall CVs were 6.0%, 6.3%, 1.6%, and 3.1% for the Elliptical method, the Visitrak device, the TeleDiaFoS system, and the SilhouetteMobile device, respectively. The precision of the Visitrak device was revealed to be remarkably lower for small wounds (<2 cm²). The Elliptical method gives overestimation up to 33%; thus, it should not be used in applications where the actual wound area is an important parameter (like the prediction of wound healing). The TeleDiaFoS system and the SilhouetteMobile device showed the best accuracy of all used methods; however, the precision of the TeleDiaFoS system was revealed to be higher than the precision of the SilhouetteMobile device. The accuracy and the precision of the Visitrak device are significantly reduced for wounds smaller than 2 cm².

Regularity of the Maxwell equations in heterogeneous media and Lipschitz domains

This note establishes regularity estimates for the solution of the Maxwell equations in Lipschitz domains with non-smooth coefficients and minimal regularity assumptions. The argumentation relies on elliptic regularity estimates for the Poisson problem with non-smooth coefficients.

An approach to improve Rayleigh-wave ellipticity estimates from seismic noise: application to the Los Angeles Basin

S U M M A R Y We present an approach for seismic noise which improves Rayleigh-wave ellipticity estimates by reducing the influences from body waves and Love waves. The method requires three-component seismographs and uses the phase-shift information between the vertical and horizontal components. We select data that show the phase-shift of about 90◦ in order to separate Rayleigh waves from body waves and Love waves. In comparison to the usual H/V approach, the estimates for Rayleigh-wave ellipticity are systematically different by about 20 per cent because the existence of S waves and Love waves systematically increases horizontal amplitudes. The differences in the inverted S-wave velocity structure reach up to about 20 per cent in the upper 20 km. The method is limited to a relatively low-frequency range, below 0.3 Hz, as the 90◦ phase-shift peaks become difficult to identify above 0.3 Hz. Despite this limitation in the frequency range, this approach can improve the S-wave velocity structure in the upper 5–10 km, which should lead to a better prediction of long-period ground motions for periods of about 5–10 s. An improved prediction of such long-period ground motion is important for hazard mitigation in the world’s metropolitan areas with high-rise buildings. This approach is applied to the structure in the broader Los Angeles basin in Southern California and improvement of the reference velocity structure in the upper 5–10 km is demonstrated.

Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces

We define a very general “parametric connect sum” construction which can be used to eliminate isolated conical singularities on Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used in the construction. Specifically, we prove uniform estimates related to (i) Sobolev Embedding Theorems, (ii) the invertibility of the Laplace operator and (iii) Poincare and Gagliardo-Nirenberg-Sobolev type inequalities. For a geometric application of these results we refer the reader to our paper [15] concerning desingularizations of special Lagrangian conifolds in C.

Range Prediction and Trajectory Correction of Long Range Rocket with Attitude Stabilization

The dispersion of long range rocket is significantly suppressed by the attitude stabilization in the boost phase. However, the attitude stabilization system cannot govern the propellant impulse error and mass error, which induces large dispersion in the longitudinal direction. Therefore, it is necessary to conduct range correction in the post-boost phase of flight trajectory. A range prediction method based on the elliptic ballistic theory is proposed in this paper. Elliptic ballistic equations and range estimation equations are derived and the modified range prediction method with correction factor is presented. It is verified that the proposed method can predict the residual flight range accurately. After that, the lateral pulse jets control is presented and utilized to correct the trajectory and eliminate the range error. A unique control law is reported that combines elliptic ballistic theory and control mode for lateral pulse jets. The impact point is directly controlled and rocket dispersion in the longitudinal direction is efficiently reduced.

A simple proof of 𝐿^{𝑞}-estimates for the steady-state Oseen and Stokes equations in a rotating frame. Part I: Strong solutions

Consider a rigid body moving in a three-dimensional Navier-Stokes liquid with a prescribed velocity ξ ∈ R 3 \xi \in \mathbb {R}^3 and a non-zero angular velocity ω ∈ R 3 ∖ { 0 } \omega \in \mathbb {R}^3\setminus \{ 0\} that are constant when referred to a frame attached to the body. By linearizing the associated equations of motion, we obtain the Oseen ( ξ ≠ 0 \xi \neq 0 ) or Stokes ( ξ = 0 \xi =0 ) equations in a rotating frame of reference. We will consider the corresponding steady-state whole-space problem. Our main result in this first part concerns elliptic estimates of the solutions in terms of data in L q ( R 3 ) L^{q}(\mathbb {R}^3) . Such estimates have been established by R. Farwig in Tohoku Math. J., Vol. 58, 2006, for the Oseen case, and R. Farwig, T. Hishida, and D. Müller in Pacific J. Math., Vol. 215 (2), 2004, for the Stokes case. We introduce a new approach resulting in an elementary proof of these estimates. Moreover, our method yields more details on how the constants in the estimates depend on ξ \xi and ω \omega . In part II we will establish similar estimates in terms of data in the negative order homogeneous Sobolev space D 0 − 1 , q ( R 3 ) D^{-1,q}_0(\mathbb {R}^3) .