Interior Estimate Research Articles

An Interior Maximum Norm Error Estimate for the Symmetric Interior Penalty Method on Planar Polygonal Domains

Abstract We establish an interior maximum norm error estimate for the symmetric interior penalty method on planar polygonal domains.

Tentative Evidence for Water Vapor in the Atmosphere of the Neptune-sized Exoplanet HD 106315c

We present a transmission spectrum for the Neptune-sized exoplanet HD 106315c from optical to infrared wavelengths based on transit observations from the Hubble Space Telescope/Wide Field Camera 3, K2, and Spitzer. The spectrum shows tentative evidence for a water absorption feature in the 1.1–1.7 μm wavelength range with a small amplitude of 30 ppm (corresponding to just 0.8 ± 0.04 atmospheric scale heights). Based on an atmospheric retrieval analysis, the presence of water vapor is tentatively favored with a Bayes factor of 1.7–2.6 (depending on prior assumptions). The spectrum is most consistent with either an enhanced metallicity or high-altitude condensates, or both. Cloud-free solar composition atmospheres are ruled out at >5σ confidence. We compare the spectrum to grids of cloudy and hazy forward models and find that the spectrum is fit well by models with moderate cloud lofting or haze formation efficiency over a wide range of metallicities (1–100× solar). We combine the constraints on the envelope composition with an interior structure model and estimate that the core mass fraction is ≳0.3. With a bulk composition reminiscent of that of Neptune and an orbital distance of 0.15 au, HD 106315c hints that planets may form out of broadly similar material and arrive at vastly different orbits later in their evolution.

Global gradient estimates for nonlinear parabolic operators

We consider a parabolic equation driven by a nonlinear diffusive operator and we obtain a gradient estimate in the domain where the equation takes place. This estimate depends on the structural constants of the equation, on the geometry of the ambient space and on the initial and boundary data. As a byproduct, one easily obtains a universal interior estimate, not depending on the parabolic data. The setting taken into account includes sourcing terms and general diffusion coefficients. The results are new, to the best of our knowledge, even in the Euclidean setting, though we treat here also the case of a complete Riemannian manifold.

Interior C2 regularity of convex solutions to prescribing scalar curvature equations

We establish interior $C^2$ estimates for convex solutions of scalar curvature equation and $\sigma_2$-Hessian equation. We also prove interior curvature estimate for isometrically immersed hypersurfaces $(M^n,g)\subset \mathbb R^{n+1}$ with positive scalar curvature. These estimates are consequences of an interior estimate for these equations obtained under a weakened condition.

Affine techniques on extremal metrics on toric surfaces

This paper consists of a package of real and complex affine techniques for the Abreu equation on toric surfaces/manifolds. In particular, as an application of this package we give an interior estimate for the Ricci tensor of toric surfaces.

Parabolic Biased Infinity Laplacian Equation Related to the Biased Tug-of-War

Abstract In this paper, we study the parabolic inhomogeneous β-biased infinity Laplacian equation arising from the β-biased tug-of-war u t - Δ ∞ β ⁢ u = f ⁢ ( x , t ) , {u_{t}}-\Delta_{\infty}^{\beta}u=f(x,t), where β is a fixed constant and Δ ∞ β {\Delta_{\infty}^{\beta}} is the β-biased infinity Laplacian operator Δ ∞ β ⁢ u = Δ ∞ N ⁢ u + β ⁢ | D ⁢ u | \Delta_{\infty}^{\beta}u=\Delta_{\infty}^{N}u+\beta\lvert Du\rvert related to the game theory named β-biased tug-of-war. We first establish a comparison principle of viscosity solutions when the inhomogeneous term f does not change its sign. Based on the comparison principle, the uniqueness of viscosity solutions of the Cauchy–Dirichlet boundary problem and some stability results are obtained. Then the existence of viscosity solutions of the corresponding Cauchy–Dirichlet problem is established by a regularized approximation method when the inhomogeneous term is constant. We also obtain an interior gradient estimate of the viscosity solutions by Bernstein’s method. This means that when f is Lipschitz continuous, a viscosity solution u is also Lipschitz in both the time variable t and the space variable x. Finally, when f = 0 {f=0} , we show some explicit solutions.

The growth of gradients of solutions of some elliptic equations and Bi-Lipschicity of QCH

In this paper, we study the growth of gradients of solutions of elliptic equations, including the Dirichlet eigenfunction solutions on bounded plane convex domain. Several results related to Bi-Lipschicity of quasiconformal harmonic (qch) mappings with respect to quasi-hyperbolic and euclidean metrics, are proved. In connection with the subject, we announce a few results concerning the so called interior estimate, including Proposition 1.1. In addition, a short review of the subject is given.

Sufficient Conditions for First-Order Differential Operators to be Associated with a q-Metamonogenic Operator in a Clifford Type Algebra

Consider the initial value problem 0.1 $$\begin{aligned} \partial _{t}u= & {} {\mathcal L}(t,x,u,\partial _{x_{i}}u),\nonumber \\ u(0,x)= & {} \varphi (x), \end{aligned}$$ where t is the time, $${\mathcal L}$$ is a linear first-order differential operator and $$\varphi $$ is a generalized q-metamonogenic function. This problem can be solved by applying the method of associated spaces which is constructed by Tutschke (see Solution of initial value problems in classes of generalized analytic functions, Teubner Leipzig and Springer, New York, 1989). In this work, we formulate sufficient conditions on the coefficients of the operator $${\mathcal L}$$ under which this operator is associated to the space of generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side, when q and $$\lambda $$ are constant Clifford vectors. We also build a computational algorithm to check the computations in the cases $${\mathcal A}^{*}_{2,2}$$ and $${\mathcal A}^{*}_{3,2}$$ . In conical domains, the initial value problem (0.1) is uniquely solvable for an operator $${\mathcal L}$$ and for any generalized q-metamonogenic initial function $$\varphi $$ , provided an interior estimate holds for generalized q-metamonogenic functions satisfying a differential equation with anti-q-metamonogenic right-hand side. The solution is also a generalized q-metamonogenic function for each fixed t. This work generalizes the results given in Di Teodoro and Sapian (Adv. Appl. Clifford Algebras, 25:283–301, 2015) and Van (Differential operator in a Clifford analysis associated to differential equations with anti-monogenic right hand side, IC/2006/134, 2016).

The interior regularity of the Calabi flow on a toric surface

Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for \(t < T\).

An interior gradient estimate for the mean curvature equation of Killing graphs and applications

We extend the interior gradient estimate due to Korevaar-Simon for solutions of the mean curvature equation from the case of euclidean graphs to the general case of Killing graphs. Our main application is the proof of existence of Killing graphs with prescribed mean curvature function for continuous boundary data, thus extending a result due to Dajczer, Hinojosa, and Lira. In addition, we prove the existence and uniqueness of radial graphs in hyperbolic space with prescribed mean curvature function and asymptotic boundary data at infinity.

An interior estimate for convex solutions and a rigidity theorem

We establish an interior C 2 estimate for k + 1 convex solutions to Dirichlet problems of k -Hessian equations. We also use such estimate to obtain a rigidity theorem for k + 1 convex entire solutions of k -Hessian equations in Euclidean space.

Hölder stability estimate of Robin coefficient in corrosion detection with a single boundary measurement

This paper is concerned with the inverse problem of detecting a boundary corrosion coefficient which describes some corrosion index from a single pair of Cauchy data measured on an accessible boundary of an electrostatic conductor in two dimensions. The corroded portion is supposed to be either a line segment or a part of some circle, while the corrosion coefficient is restricted to be an analytic or a piecewise constant function. We prove two Hölder stability estimates in recovering the unknown boundary coefficient. Our arguments rely on the Schwarz reflection principle with the Robin boundary condition and a novel interior estimate derived from the elliptic Carleman estimate.

Interior estimates in sup-norm for generalized potential vectors

The initial value problem of the type can be solved by contraction mapping principle in the case that the initial function ϕ belongs to an associated space, whose elements satisfy an interior estimate. This article proves such interior estimate in the sup-norm for the generalized potential vectors. This proof is based on representation of generalized potential vectors by potential vectors. In that way this article is another example for the technique of transforming solutions of simpler partial differential equations into solutions of more general ones (see H. Begehr and R. P. Gilbert [4]).

Initial Value Problems in Quaternionic Analysis with a Disturbed Dirac Operator

Initial value problems (IVPs) of type $$\frac{\partial u}{\partial t} = L ( t, x, u, \frac{\partial u}{\partial x_j} ), \quad u(0, x) = \varphi (x)$$ can be solved by applying the method of associated spaces which is constructed by W.Tutschke (Teubner Leipzig and Springer-Verlag 1989). The present paper considers above IVPs in the space of Helmholtz-type generalized regular functions in the sense of quaternionic analysis. Using the Poisson integral formula, we shall prove an interior estimate for Helmholtz-type generalized regular functions and then give out conditions under which these IVPs are uniquely solvable.

Regression Model for Interior Design Cost Estimate in Preliminary Stage

Cost factors become even more important in the design of the kitchen which is remodeled 34% more than any other room of the house. For this reason, the purpose of this study was to propose a reliable kitchen cost estimate model that can be used during the pre-design stages. The first stage of the methodology consisted of defining the limits and the parameters of the model. Next, 1.309 kitchen design projects were analyzed for data and a regression model based on the correlations between these data was developed. In the last stage, sample cases were developed to prove the feasibility of using the kitchen remodeling cost estimate model.

General curvature estimates for stable H-surfaces in 3-manifolds applications

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.

On the Evans-Krylov theorem

We provide a short proof of the C 2,α interior estimate for convex fully nonlinear elliptic equations. This result was originally proved by L. C. Evans and N. Krylov. Our proof is based on the ideas from our work on integro-differential equations.

Phragmén–Lindelöf type theorem for m-hessian equations

We derive an interior estimate for the gradient of a solution u to the m-Hessian equation with a certain right-hand side. The estimate depends on the oscillation of u and properties of the right-hand side of the equation. The proof is based on a modification of some ideas of Trudinger (1997). As a consequence of the main result, a theorem of Fragmen–Lindelof type is obtained for solutions to the m-Hessian equations in the entire space $$ \mathbb{R}^n $$ . Bibliography: 4 titles.

Interior estimates in the sup-norm for generalized monogenic functions satisfying a differential equation with an anti-monogenic right-hand side

Initial value problems of type can be solved by the contraction-mapping principle in case the initial function ϕ belongs to an associated space whose elements satisfy an interior estimate. The present article proves such an interior estimate in the sup-norm for generalized monogenic functions. The proof is based on a representation of generalized monogenic functions by harmonic functions. That way the article is another example for the technique of transforming solutions of simpler partial differential equations into solutions of more general ones (an overview on such methods can be found in Begehr's and Gilbert's two volumes [Begehr, H.G.W. and Gilbert, R.P., 1992/1993, Transformations, Transmutations and Kernel Functions, Vol. I and II (Harlow: Longman Scientific & Technical; New York: John Wiley & Sons, Inc.)].

Uniform Elliptic Estimate for an Infinite Plate in Linear Elasticity

We present a new study of linear elasticity for an infinite three-dimensional plate of finite thickness Ω = ℝ2 × (−1, 1). We first characterize the kernel of the operator of elasticity as polynomials which can be build from the kernel of the classical Kirchhoff–Love model of plate. Using this characterization, we get optimal uniform elliptic estimates W k, p , C k, α on the solution as a function of the exterior forces. We also give an interior estimate.