Uniform Ellipticity Research Articles

An approximate maximum likelihood estimator of drift parameters in a multidimensional diffusion model

For a fixed \(T\) and \(k \geq 2\), a \(k\)-dimensional vector stochastic differential equation \(dX_t=\mu(X_t, \theta)\,dt+\nu(X_t)\,dW_t,\) is studied over a time interval \([0,T]\). Vector of drift parameters \(\theta\) is unknown. The dependence in \(\theta\) is in general nonlinear. We prove that the difference between approximate maximum likelihood estimator of the drift parameter \(\overline{\theta}_n\equiv \overline{\theta}_{n,T}\) obtained from discrete observations \((X_{i\Delta_n}, 0 \leq i \leq n)\) and maximum likelihood estimator \(\hat{\theta}\equiv \hat{\theta}_T\) obtained from continuous observations \((X_t, 0\leq t\leq T)\), when \(\Delta_n=T/n\) tends to zero, converges stably in law to the mixed normal random vector with covariance matrix that depends on \(\hat{\theta}\) and on path \((X_t, 0 \leq t\leq T)\). The uniform ellipticity of diffusion matrix \(S(x)=\nu(x)\nu(x)^T\) emerges as the main assumption on the diffusion coefficient function.

Unconditionally optimal error estimates of linearized Crank-Nicolson virtual element methods for quasilinear parabolic problems on general polygonal meshes

In this paper, we construct, analyze, and numerically validate a linearized Crank-Nicolson virtual element method (VEM) for solving quasilinear parabolic problems on general polygonal meshes. In particular, we consider the more general nonlinear term a(x, u), which does not require Lipschitz continuity or uniform ellipticity conditions. To ensure that the fully discrete solution remains bounded in L∞-norm, we construct two novel elliptic projections and apply a new error splitting technique. With the help of the boundedness of numerical solution and delicate analysis of the nonlinear term, we derive the optimal error estimates for any k-order VEMs without any time-step restrictions. Numerical experiments on various polygonal meshes validate the accuracy of theoretical analysis and the unconditional convergence of the proposed scheme.

On the Dirichlet problem for beltrami equations with sources in simply connected domains

In this paper, we present our recent results on the solvability of the Dirichlet problem Reω(z) → φ (ζ) as z → ζ, z∈ D, ζ∈ ∂D, with continuous boundary data φ: ∂D ???? R for degenerate Beltrami equations ωz =µ(z)ω2 + σ(z), |µ(z) ˂ 1 a.e., with sources σ: D → C that belong to the class Lp (D), p ˃ 2, and have compact supports in D. In the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class W1.2loc (D) of the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.

Exterior Boundary-Value Poincaré Problem for Elliptic Systems of the Second Order with Two Independent Variables

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether.Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether.This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid.

On divergence type linear and quasilinear equations in the complex plane

This is a survay of our recent results concerning divergence type linear and quasilinear equations in the complex plane. It contains a number of existence, representation and regularity theorems for solutions of fundamental boundary value problems for such equations. The degeneration case of uniform ellipticity is also covered by means of BMO, VMO, Calderon-Zigmund, Lehto and Orlicz technic.

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form

An augmented mixed FEM for the convective Brinkman–Forchheimer problem: a priori and a posteriori error analysis

We propose and analyze an augmented mixed finite element method for the pseudostress–velocity formulation of the stationary convective Brinkman–Forchheimer problem in Rd,d∈{2,3}. Since the convective and Forchheimer terms forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms. The resulting augmented scheme is written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with the Lax–Milgram theorem, allow to prove the unique solvability of the continuous problem. The finite element discretization involves Raviart–Thomas spaces of order k≥0 for the pseudostress tensor and continuous piecewise polynomials of degree ≤k+1 for the velocity. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. In addition, we derive two reliable and efficient residual-based a posteriori error estimators for this problem on arbitrary polygonal and polyhedral regions. The reliability of the proposed estimators draws mainly upon the uniform ellipticity of the form involved, a suitable assumption on the data, a stable Helmholtz decomposition, and the local approximation properties of the Clément and Raviart–Thomas operators. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, some numerical examples illustrating the performance of the mixed finite element method, confirming the theoretical rate of convergence and the properties of the estimators, and showing the behavior of the associated adaptive algorithms, are reported. In particular, the case of flow through a 2D porous media with fracture networks is considered.

L2 error estimates for a family of cubic finite volume methods on triangular meshes

This paper considers the error estimates of a family of Lagrange cubic finite volume methods for solving two dimensional elliptic boundary value problems over triangular meshes. By introducing a novel map of finite volume method, we propose a family of cubic finite volume method schemes with four independent map coefficients. It is detailedly proved that the proposed finite volume method schemes is exactly equivalent to the existing traditional finite volume method schemes of the same boundary value problems. A geometric constraint requirement on the triangular meshes of primary partitions is imposed to guarantee the uniform ellipticity of discrete elliptic bilinear forms resulted from the proposed schemes. The geometric requirements of the primary partitions could be greatly attenuated when the map coefficients are appropriately designed. Once the uniform ellipticity of the discrete elliptic bilinear forms is established, the proposed schemes have the unique solvability, and the optimal H1-norm error estimates without any restriction on dual elements of dual partitions of the finite volume methods, which is consistent with the results of literatures. With the help of the introduced map, the proposed schemes have the optimal L2-norm error estimates with a mesh restriction for the dual elements of their dual partitions significantly weaker than the existing theoretical results, in which the Aubin-Nitsche duality argument and the orthogonal conditions based on the dual partitions are employed. Finally, we validate our theoretical results by numerical experiments.

The Dirichlet problem for the Beltrami equations with sources

The paper is devoted to the study of the Dirichlet problem Re ω(z) → φ(ζ) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ : ∂D → ℝ for Beltrami equations $$ {\omega}_{\overline{z}} $$ = μ(z)ωz + σ(z), |μ(z)| < 1 a.e., with sources σ : D → ℂ in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains D of the complex plane ℂ with no boundary component degenerated to a single point for sources σ in Lp (D), p > 2, with compact support in D. Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation div[A(z)∇ u(z)] = g(z), whose source g ∈ Lp(D), p > 1, has compact support in D and whose matrix-valued coefficient A(z) guarantees its locally uniform ellipticity.

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.

The Dirichlet problem for the Beltrami equations with sources

The paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,\omega(z)\to\varphi(\zeta)$ as $z\to\zeta,$ $z\in D,\zeta\in \partial D,$ with continuous boundary data $\varphi :\partial D\to\mathbb R$ for Beltrami equations $\omega_{\bar{z}}=\mu(z) \omega_z+\sigma (z)$, $|\mu(z)|&lt;1$ a.e., with sources $\sigma :D\to\mathbb C$ in the case of locally uniform ellipticity. In this case, we have established a series of effective integral criteria of the BMO, FMO, Calderon-Zygmund, Lehto, and Orlicz types on the singularities of the equations at the boundary for the existence, representation, and regularity of solutions in arbitrary bounded domains $D$ of the complex plane $\mathbb C$ with no boun\-da\-ry component degenerated to a single point for sources $\sigma$ in $L_p(D)$, $p&gt;2$, with compact support in $D$. Moreover, we have proved the existence, representation, and regularity of weak solutions of the Dirichlet problem in such domains for the Poisson-type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$, whose source $g\in L_p(D)$, $p&gt;1$, has compact support in $D$ and whose mat\-rix-valued coefficient $A(z)$ guarantees its locally uniform ellipticity.

Unadjusted Langevin algorithm with multiplicative noise: Total variation and Wasserstein bounds

In this paper, we focus on nonasymptotic bounds related to the Euler scheme of an ergodic diffusion with a possibly multiplicative diffusion term (nonconstant diffusion coefficient). More precisely, the objective of this paper is to control the distance of the standard Euler scheme with decreasing step (usually called unadjusted Langevin algorithm in the Monte Carlo literature) to the invariant distribution of such an ergodic diffusion. In an appropriate Lyapunov setting and under uniform ellipticity assumptions on the diffusion coefficient, we establish (or improve) such bounds for total variation and L1-Wasserstein distances in both multiplicative and additive and frameworks. These bounds rely on weak error expansions using stochastic analysis adapted to decreasing step setting.

A weak solution to a perturbed one-Laplace system by p-Laplacian is continuously differentiable

In this paper we aim to show continuous differentiability of weak solutions to a one-Laplace system perturbed by p-Laplacian with \(1<p<\infty \). The main difficulty on this equation is that uniform ellipticity breaks near a facet, the place where a gradient vanishes. We would like to prove that derivatives of weak solutions are continuous even across the facets. This is possible by estimating Hölder continuity of the Jacobian matrix multiplied with its modulus truncated near zero. To show this estimate, we consider an approximated system, and use standard methods including De Giorgi’s truncation and freezing coefficient arguments.

Precise Local Estimates for Differential Equations driven by Fractional Brownian Motion: Elliptic Case

This article is concerned with stochastic differential equations driven by a d-dimensional fractional Brownian motion with Hurst parameter \(H>1/4\), understood in the rough paths sense. Whenever the coefficients of the equation satisfy a uniform ellipticity condition, we establish a sharp local estimate on the associated control distance function and a sharp local lower estimate on the density of the solution.

Boundary differentiability of solutions to elliptic equations in convex domains in the borderline case

In this work, we consider the following elliptic partial differential equations: $$\begin{aligned} \left\{ \begin{aligned} - b_{ij} \, \frac{\partial ^{2} w}{\partial x_{i} \partial x_{j}}&= g \,\,\, \text {in} \,\, \Omega , \\ w&= 0 \,\,\,\text {on} \,\partial \Omega , \end{aligned} \right\} . \end{aligned}$$where the domain \(\Omega \subset {\mathbb {R}}^{n}\) is convex, the matrix \(\big (b_{ij}\big )_{n \times n}\) satisfies the uniform ellipticity conditions. For g in the scaling critical Lorentz space \( L(n,\, 1)(\Omega )\), we establish boundary differentiability of solutions to the above problem. We also prove \(C^{\mathrm {Log-Lip}}\) regularity estimate at a boundary point in the case when \(g \in L^{n}(\Omega )\).

Approximation of a degenerate semilinear PDE with a nonlinear Neumann boundary condition

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required for the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required for the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is monotone in the solution variable. Note that the existence of a viscosity solution to this problem has been established in [13] then completed in [15]. In the present paper, we construct a sequence of penalized systems of decoupled forward backward stochastic differential equations (FBSDEs) then we directly show its strong convergence. This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1, 3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem.

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

&lt;p style='text-indent:20px;'&gt;We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotonicity of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems at &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ x = \pm \infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Most notably, when the profile of the diffusion coefficient is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the diffusion coefficient is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.&lt;/p&gt;

SLLN and annealed CLT for random walks in I.I.D. random environment on Cayley trees

We consider the random walk in an independent and identically distributed (i.i.d.) random environment on a Cayley graph of a finite free product of copies of Z and Z2. Such a Cayley graph is readily seen to be a regular tree. Under a uniform ellipticity assumption on the i.i.d. environment we show that the walk has positive speed and establish the annealed central limit theorem for the graph distance of the walker from the starting point.

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.

Lipschitz minimizers for a class of integral functionals under the bounded slope condition

We consider the functional ∫Ωg(∇u+X∗)dL2n where g is convex and X∗(x,y)=2(−y,x) and we study the minimizers in BV(Ω) of the associated Dirichlet problem. We prove that, under the bounded slope condition on the boundary datum, and suitable conditions on g, there exists a unique minimizer which is also Lipschitz continuous. The assumptions on g allow to consider both the case with superlinear growth and the one with linear growth. Moreover neither uniform ellipticity nor smoothness of g are assumed.