Uniformly Elliptic Research Articles

Reduced order modeling for elliptic problems with high contrast diffusion coefficients

We consider a parametric elliptic PDE with a scalar piecewise constant diffusion coefficient taking arbitrary positive values on fixed subdomains. This problem is not uniformly elliptic, as the contrast can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is com- monly made on parametric elliptic PDEs. We construct reduced model spaces that approximate uniformly well all solutions with estimates in relative error that are independent of the contrast level. These estimates are sub-exponential in the reduced model dimension, yet exhibiting the curse of dimensionality as the number of subdomains grows. Similar es- timates are obtained for the Galerkin projection, as well as for the state estimation and parameter estimation inverse problems. A key ingredient in our construction and analysis is the study of the convergence towards limit solutions of stiff problems when diffusion tends to infinity in certain domains.

Finitely presented nilsemigroups: complexes with the property of uniform ellipticity

This paper is the first in a series of three devoted to constructing a finitely presented infinite nilsemigroup satisfying the identity . This solves a problem of Lev Shevrin and Mark Sapir. In this first part we obtain a sequence of complexes formed of squares (-cycles) having the following geometric properties. 1) Complexes are uniformly elliptic. A space is said to be uniformly elliptic if there is a constant such that in the set of shortest paths of length connecting points and there are two paths such that the distance between them is at most . In this case, the distance between paths with the same beginning and end is defined as the maximal distance between the corresponding points. 2) Complexes are nested. A complex of level is obtained from a complex of level by adding several vertices and edges according to certain rules. 3) Paths admit local transformations. Assume that we can transform paths by replacing a path along two sides of a minimal square by the path along the other two sides. Two shortest paths with the same ends can be transformed into each other locally if these ends are vertices of a square in the embedded complex. The geometric properties of the sequence of complexes will be further used to define finitely presented semigroups.

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on subsolutions of the equation − A v + c ( x ) v = 0 , x ∈ D , where D ⊂ R d is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered “quantitative” versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

Optimal Homogenization Rates in Stochastic Homogenization of Nonlinear Uniformly Elliptic Equations and Systems

We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale varepsilon >0, we establish homogenization error estimates of the order varepsilon in case dgeqq 3, and of the order varepsilon |log varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence varepsilon ^delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,alpha } regularity theory is available.

Law of Large Numbers for Random Walk with Unbounded Jumps and Birth and Death Process with Bounded Jumps in Random Environment

We study a random walk with unbounded jumps in random environment. The environment is stationary and ergodic, uniformly elliptic and decays polynomially with speed \(Dj^{-(3+\varepsilon _0)}\) for some \(D>0\) and small \(\varepsilon _0>0.\) We prove a law of large numbers under the condition that the annealed mean of the hitting time of the lattice of the positive half line is finite. As the second part, we consider a birth and death process with bounded jumps in stationary and ergodic environment whose skeleton process is a random walk with unbounded jumps in random environment. Under a uniform ellipticity condition, we prove a law of large numbers and give the explicit formula of its velocity.

The Dirichlet problem with prescribed interior singularities

In this paper we solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in Rn. The main results apply, in particular, to subequations with a Riesz characteristic p≥2. It is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel ΘjKp(x−xj) whereKp(x)={−1|x|p−2for2<p<∞,log⁡|x|ifp=2. at any prescribed finite set of points {x1,...,xk} in the domain and any finite set of positive real numbers Θ1,...,Θk. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions.Uniqueness and existence results are also established for finite-type singularities such as Θj|x−xj|2−p for 1≤p<2.The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong–Sirakov–Smart (in the uniformly elliptic case).

Error estimates for approximations of nonhomogeneous nonlinear uniformly elliptic equations

We obtain an error estimate between viscosity solutions and \delta-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is Lipschitz-continuous in space with linear growth in the Hessian. We also establish a rate of convergence for monotone and consistent finite difference approximation schemes for such equations.

Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity

We prove regularity and stochastic homogenization results for certain degenerate elliptic equations in nondivergence form. The equation is required to be strictly elliptic, but the ellipticity may oscillate on the microscopic scale and is only assumed to have a finite $d$th moment, where $d$ is the dimension. In the general stationary-ergodic framework, we show that the equation homogenizes to a deterministic, uniformly elliptic equation, and we obtain an explicit estimate of the effective ellipticity, which is new even in the uniformly elliptic context. Showing that such an equation behaves like a uniformly elliptic equation requires a novel reworking of the regularity theory. We prove deterministic estimates depending on averaged quantities involving the distribution of the ellipticity, which are controlled in the macroscopic limit by the ergodic theorem. We show that the moment condition is sharp by giving an explicit example of an equation whose ellipticity has a finite $p$th moment, for every $p<d$, but for which regularity and homogenization break down. In probabilistic terms, the homogenization results correspond to quenched invariance principles for diffusion processes in random media, including linear diffusions as well as diffusions controlled by one controller or two competing players.

General Boundary Value Problems for Nonlinear Uniformly Elliptic Equations in Multiply Connected Infinite Domains

This article discusses the general boundary value problem for the nonlinear uniformly elliptic equation of second order in D (0.1) and the boundary condition,(0.2) in a multiply connected infinite domain D with the boundary T. The above boundary value problem is called Problem G. Problem G extends the work [8] in which the equation (0.1) includes a nonlinear lower term and the boundary condition (0.2) is more general. If the complex equation (0.1) and the boundary condition (0.2) meet certain assumptions, some solvability results for Problem G can be obtained. By using reduction to absurdity, we first discuss a priori estimates of solutions and solvability for a modified problem. Then we present results on solvability of Problem G.

The probabilistic Brosamler formula for some nonlinear Neumann boundary value problems governed by elliptic possibly degenerate operators

This paper concerns with boundary value problems as { L u+a0u = f in Ω, +c0|u|m−1u = g on ∂Ω, where L is an elliptic possibly degenerate second order operator, a0, c0 are positive function, γ is an oblique exterior vector and m 1 . By means of some arguments close to the Dynamics Programming we prove that the viscosity solution admits a representation formula that can be considered as an extension of probabilistic Brosamler formula of linear Neumann boundary value problems governed by uniformly elliptic operators. Although other generalizations are possible, by simplicity we limit this contribution to the presence of nonlinear terms exclusively on the boundary of the domain. We emphasize that any uniforme ellipticity assumption is required in the paper.

Viscosity Solutions of Uniformly Elliptic Equations without Boundary and Growth Conditions at Infinity

We deal with fully nonlinear second‐order equations assuming a superlinear growth in u with the aim to generalize previous existence and uniqueness results of viscosity solutions in the whole space without conditions at infinity. We also consider the solvability of the Dirichlet problem in bounded and unbounded domains and show a blow‐up result.

Solvability of Uniformly Elliptic Fully Nonlinear PDE

We get existence, uniqueness and non-uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the Hamilton–Jacobi–Bellman–Isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness. Some of our results are new even for equations in divergence form.

On a class of hypoelliptic operators with unbounded coefficients in $R^N$

We consider second order linear partial differential operators $A$ on $R^N$ which are not assumed to be uniformly elliptic and whose coefficients in the second order part may grow quadratically, while the drift part has essentially linear growth. Instead of uniform ellipticity, we require a much weaker hypothesis of uniform hypoellipticity, which in an equivalent formulation connects the behaviour of the diffusion and the drift coefficients, by requiring that a Kalman-type condition is satisfied for them. By refining Bernstein's method we prove the existence of a semigroup {$T(t)$} of bounded linear operators (in the space of bounded and continuous functions) associated to the operator $A$. We also show uniform estimates for the spatial derivatives of the semigroup {$T(t)$} in (an)isotropic spaces of (Holder-) continuous functions. As a consequence, we obtain Holder estimates for the solutions of some elliptic and parabolic problems associated to the operator $A$.

An Estimate on the Heat Kernel of Magnetic Schrödinger Operators and Uniformly Elliptic Operators with Non-Negative Potentials

The paper studies the heat kernel of the Schrödinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Hölder class which includes non-negative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.

On the Lp-Spectrum of Uniformly Elliptic Operators on Riemannian Manifolds

We prove that the Lp spectrum of uniformly elliptic divergence form operators on a complete Riemannian manifold is independent of p ∈ [1, ∞] if the volume of the manifold grows uniformly subexponentially. The latter condition is satisfied if the Ricci curvature is bounded from below by a function K: M → R satisfying lim infx → ∞K(x) ≥ 0, in particular, if the Ricci curvature is nonnegative. On the other hand, if we asume the existence of a (globally defined) volume density which grows exponentially in every direction, then the Lp spectrum depends on p. This condition is satisfied if the manifold is simply connected and if the sectional curvature is bounded from above by a function K: M → R satisfying K ≤ 0 and lim supx → ∞K(x) < 0. For instance, it is satisfied for the hyperbolic space.

Estimates on the Fundamental Solution to Heat Flows With Uniformly Elliptic Coefficients

We obtain estimates, from above and below, on the fundamental solution to the Cauchy initial value problem for parabolic equations having uniformly elliptic coefficients. We impose minimal regularity requirements on the coefficients. The estimates are expressed in terms of an energy functional associated with the coefficients

Relative influence of two small parameters in non uniformly elliptic homogenization problems

AbstractWe study three elliptic problems depending on two small parameters (ϵ = homogenization parameter and δ = perturbation parameter which causes non uniform ellipticity). In each case, the homogenized operator corresponding to the second order operator is independent of the way (ϵ, δ) → (0, 0), but the convergence results and the limit solution do depend on the relative size of ϵ and δ; the relevant parameter is δϵ−2.

On the Neumann Problem for some Linear and Non-linear Uniformly Elliptic Difference Equations

On the Neumann Problem for some Linear and Non-linear Uniformly Elliptic Difference Equations

Quasilinear Uniformly Elliptic Partial Differential Equations and Difference Equations

Previous article Next article Quasilinear Uniformly Elliptic Partial Differential Equations and Difference EquationsG. I. McAllisterG. I. McAllisterhttps://doi.org/10.1137/0703002PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Lipman Bers, On mildly nonlinear partial difference equations of elliptic type, J. Research Nat. Bur. Standards, 51 (1953), 229–236 MR0064291 0053.40202 CrossrefISIGoogle Scholar[2] J. H. Bramble and , B. E. Hubbard, On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation, Numer. Math., 4 (1962), 313–327 10.1007/BF01386325 MR0149672 0135.18102 CrossrefGoogle Scholar[3] Lothar Collatz, The numerical treatment of differential equations. 3d ed, Translated from a supplemented version of the 2d German edition by P. G. Williams. Die Grundlehren der mathematischen Wissenschaften, Bd. 60, Springer-Verlag, Berlin, 1960xv+568 pp. (1 plate) MR0109436 0086.32601 CrossrefGoogle Scholar[4] R. Courant and , D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, (Vol. II by R. Courant.), Interscience Publishers (a division of John Wiley & Sons), New York-Lon don, 1962xxii+830 MR0140802 0099.29504 Google Scholar[5] George E. Forsythe and , Wolfgang R. Wasow, Finite-difference methods for partial differential equations, Applied Mathematics Series, John Wiley & Sons Inc., New York, 1960x+444 MR0130124 0099.11103 Google Scholar[6] G. T. McAllister, Some nonlinear elliptic partial differential equations and difference equations, J. Soc. Indust. Appl. Math., 12 (1964), 772–777 10.1137/0112063 MR0179958 0137.08002 LinkISIGoogle Scholar[7] Tibor Radó, On the Problem of Plateau, Chelsea Publishing Co., New York, N. Y., 1951iv+109 MR0040601 Google Scholar[8] J. Schauder, Sur les équations aux dérivées partielles du type elliptique, C. R. Acad. Sci. Paris, 195 (1932), 1365–1367 0004.39502 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Monotonically convergent iterative methods for nonlinear systems of equationsNumerische Mathematik, Vol. 32, No. 1 Cross Ref On the Origins and Numerical Solution of Some Sparse Nonlinear Systems Cross Ref Velocity potential computation by finite differences for compressible flow in ductsInternational Journal for Numerical Methods in Engineering, Vol. 9, No. 3 Cross Ref Morrey space methods in the theory of elliptic difference equations23 August 2006 Cross Ref Monotonieeigenschaften Von Diskretisierungen Des Dirichletproblems Quasilinearer Elliptischer Differentialgleichungen23 August 2006 Cross Ref On equations describing steady-state carrier distributions in a semiconductor deviceCommunications on Pure and Applied Mathematics, Vol. 25, No. 6 Cross Ref Difference analogues of quasi-linear elliptic Dirichlet problems with mixed derivatives1 January 1971 | Mathematics of Computation, Vol. 25, No. 114 Cross Ref Discrete maximum principle for finite-difference operatorsAequationes Mathematicae, Vol. 4, No. 3 Cross Ref BIBLIOGRAPHY Cross Ref Existenz und Konvergenz von L�sungen nichtlinearer elliptischer Differenzengleichungen unter DirichletrandbedingungenMathematische Zeitschrift, Vol. 109, No. 4 Cross Ref An application of a priori bounds on difference quotients to a constructive solution of mildly quasilinear Dirichlet problemsJournal of Mathematical Analysis and Applications, Vol. 24, No. 3 Cross Ref A priori bounds on difference quotients of solutions to some linear uniformly elliptic difference equationsNumerische Mathematik, Vol. 11, No. 1 Cross Ref A functional analytic approach to the numerical solution of nonlinear elliptic equationsComputing, Vol. 2, No. 1 Cross Ref Volume 3, Issue 1| 1966SIAM Journal on Numerical Analysis History Submitted:23 March 1965Published online:14 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0703002Article page range:pp. 13-33ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics