Differential Equations In Divergence Form Research Articles

Singular solutions for complex second order elliptic equations and their application to time-harmonic diffuse optical tomography

We construct singular solutions of a complex elliptic equation of second order, having an isolated singularity of any order. In particular, we extend results obtained for the real partial differential equation in divergence form by Alessandrini in 1990. Our solutions can be applied to the determination of the optical properties of an anisotropic medium in time-harmonic Diffuse Optical Tomography (DOT).

Local boundedness of weak solutions to elliptic equations with $ p, q- $growth

<abstract><p>This article is dedicated to Giuseppe Mingione for his $ 50^{th} $ birthday, a leading expert in the regularity theory and in particular in the subject of this manuscript. In this paper we give conditions for the <italic>local boundedness</italic> of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type considered below in (1.1), under $ p, q- $growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on $ u $, other than on its gradient $ Du $ and on the $ x $ variable.</p></abstract>

Estimates of eigenvalues of an elliptic differential system in divergence form

In this paper, we compute universal estimates of eigenvalues of a coupled system of elliptic differential equations in divergence form on a bounded domain in Euclidean space. As an application, we show an interesting case of rigidity inequalities of the eigenvalues of the Laplacian, more precisely, we consider a countable family of bounded domains in Gaussian shrinking soliton that makes the behavior of known estimates of the eigenvalues of the Laplacian invariant by a first-order perturbation of the Laplacian. We also address the Gaussian expanding soliton case in two different settings. We finish with the special case of divergence-free tensors which is closely related to the Cheng–Yau operator.

Local Lipschitz continuity for [formula omitted]PDEs with explicit [formula omitted]dependence

This article is dedicated to Emmanuele Di Benedetto, great mathematician, colleague, friend. In the spirit to treat a subject that in the last years attracted the interest of several mathematicians, and the attention of Emmanuele too, in this paper we give a first approach to the local Lipschitz continuity of weak solutions to a class of nonlinear elliptic partial differential equations in divergence form of the type ∑i=1n∂∂xiaix,u,Du=bx,u,Duunder p,q−growth assumptions. The novelties with respect to the mathematical literature on this topic are the general growth conditions and the explicit dependence of the differential equation on u, other than on its gradient Du and on the x variable.

Scaling limit of the homogenization commutator for Gaussian coefficient fields

Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution-operator displays fluctuations around its expectation. The recently-developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension d≥1 previous results so far limited to dimension d=1, and to the continuum setting with strong correlations recent results in the discrete i.i.d. case.

New examples of Moser–Bernstein type problems for some nonlinear elliptic partial differential equations arising in geometry

A family of nonlinear partial differential equations of divergence form is considered. Each one is the Euler-Lagrange equation of a natural Riemaniann variational problem of geometric interest. New uniqueness results for the entire solutions of these equations on a parabolic Riemaniann manifold of arbitrary dimension are given. In particular, several Moser-Bernstein type theorems are proved.

Long-time asymptotics of non-degenerate non-linear diffusion equations

We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein’s random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson’s Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation’s solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

Localization of the W-1,q norm for local a posteriori efficiency

Abstract This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space ${W^{1,p}_0(\varOmega )}$, $1 \leq p \leq \infty $. The basic condition is that the functional in question vanishes over locally supported test functions from ${W^{1,p}_0(\varOmega )}$ which form a partition of unity in $\varOmega $, apart from close to the boundary $\partial \varOmega $. We also study how to weaken this condition. The results allow in particular to establish local efficiency and robustness with respect to the exponent $p$ of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers. Numerical illustrations support the theory.

Differential equations of divergence form in separable Musielak-Orlicz-Sobolev spaces

In this paper, we study the existence of weak solutions for differential equations of divergence form $$-\operatorname {div}\bigl(a_{1}(x,Du)\bigr)+a_{0}(x,u)=f(x,u,Du), $$ in Ω coupled with a Dirichlet or Neumann boundary condition in separable Musielak-Orlicz-Sobolev spaces where $a_{1}$ satisfies the growth condition, the coercive condition, and the monotone condition, and $a_{0}$ satisfies the growth condition without any coercive condition or monotone condition. The right-hand side $f:\Omega\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb {R}$ is a Caratheodory function satisfying a growth condition dependent on the solution u and its gradient Du. We prove the existence of weak solutions by using a linear functional analysis method. Some sufficient conditions guarantee the existence enclosure of weak solutions between sub- and supersolutions. Our method does not require any reflexivity of the Musielak-Orlicz-Sobolev spaces.

Uniqueness and Nonuniqueness in Inverse Problems for Elliptic Partial Differential Equations and Related Medical Imaging

Unique determination issues about inverse problems for elliptic partial differential equations in divergence form are summarized and discussed. The inverse problems include medical imaging problems including electrical impedance tomography (EIT), diffuse optical tomography (DOT), and inverse scattering problem (ISP) which is an elliptic inverse problem closely related with DOT and EIT. If the coefficient inside the divergence is isotropic, many uniqueness results are known. However, it is known that inverse problem with anisotropic coefficients has many possible coefficients giving the same measured data for the inverse problem. For anisotropic coefficient with anomaly with or without jumps from known or unknown background, nonuniqueness of the inverse problems is discussed and the relation to cloaking or illusion of the anomaly is explained. The uniqueness and nonuniqueness issues are discussed firstly for EIT and secondly for ISP in similar arguments. Arguing the relation between source-to-detector map and Dirichlet-to-Neumann map in DOT and the uniqueness and nonuniqueness of DOT are also explained.

Invariance of flows in doubly-connected domains with the same modulus

We consider systems of elliptic partial differential equations in divergence form with Dirichlet’s boundary conditions in doubly-connected domain of the plane with modulus \({\mu }\). We prove an invariance property of the corresponding global flows in the class of domains with the same modulus. Applications are given to the problem of electrical heating of a conductor whose thermal and electrical conductivities depend on the temperature and to the flow of a viscous fluid in a porous medium, taking into account the Soret and Dufour’s effects.

A priori bounds in [formula omitted] for solutions of elliptic equations in divergence form

We prove an a priori bound in Lp, p>1, for the solutions of the Dirichlet problem for second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

A Priori Bounds in and in for Solutions of Elliptic Equations

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an -a priori bound, . Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

An Lp‐Estimate for Weak Solutions of Elliptic Equations

We prove an Lp‐a priori bound, p > 2, for solutions of second order linear elliptic partial differential equations in divergence form with discontinuous coefficients in unbounded domains.

Asymptotic Expansion of the Homogenized Matrix in Two Weakly Stochastic Homogenization Settings

This article studies some numerical approximations of the homogenized matrix for stochastic linear elliptic partial differential equations in divergence form. We focus on the case when the underlying random field is a small perturbation of a reference periodic tensor. The size of such a perturbation is encoded by a real parameter eta. In this case, it has already been theoretically shown in the literature that the exact homogenized matrix possesses an expansion in powers of the parameter eta for both models considered in this article, the coefficients of which are deterministic. In practice, one cannot manipulate the exact terms of such an expansion. All objects are subjected to a discretization approach. Thus we need to derive a similar expansion for the approximated random homogenized matrix. In contrast to the expansion of the exact homogenized matrix, the expansion of the approximated homogenized matrix contains intrinsically random coefficients. In particular, the second order term is random in nature. The purpose of this work is to derive and study this expansion in function of the parameters of the approximation procedure (size of the truncated computational domain used, meshsize of the finite elements approximation).

Exact solutions of a two-fluid model of two-phase compressible flows with gravity

We aim at determining and computing a class of exact solutions of a two-fluid model of two-phase flows with/without gravity. The model is described by a non-hyperbolic system of balance laws whose characteristic fields may not be given explicitly, making it perhaps impossible to solve the Riemann problem. First, we investigate Riemann invariants in the linearly degenerate characteristic fields and obtain a surprising result on the corresponding contact waves of the model without gravity. Second, even when gravity is allowed, we show that smooth stationary solutions can be governed by a system of differential equations in divergence form, which determines jump relations for any stationary discontinuity wave. Using these relations, we establish a nonlinear equation for the pressure and propose a method to compute the pressure and then the equilibria resulted by a stationary wave.

Differential equations of divergence form in Musielak–Sobolev spaces and a sub-supersolution method

We study some properties of differential operators of divergence form in Musielak–Sobolev spaces and establish a sub-supersolution method for differential equations of divergence form with homogeneous Neumann or Dirichlet boundary condition in Musielak–Sobolev spaces.

Remarks on the existence, uniqueness and semi-explicit solvability of systems of autonomous partial differential equations in divergence form with constant boundary conditions

A method is proposed for reducing autonomous nonlinear boundary-value problems of partial differential equations with a particular structure and constant boundary conditions to the simple problemThis requires the solution of a non-standard two-point system of ordinary differential equations. Some applications are also presented for classical problems of mathematical physics.

Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem

We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in dimension $d \le 3$, from measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior random field measure on the log permeability, specified through the Karhunen–Loève expansion of its draws. We consider Gaussian measures constructed this way, and study the regularity of functions drawn from them. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Combining these regularity and continuity estimates, we show that the posterior measure is well defined on a suitable Banach space. Furthermore the posterior measure is shown to be Lipschitz with respect to the data in the Hellinger metric, giving rise to a form of well posedness of the inverse problem. Determining the posterior measure, given the data, solves the problem of uncertainty quantification for this inverse problem. In practice the posterior measure must be approximated in a finite dimensional space. We quantify the errors incurred by employing a truncated Karhunen–Loève expansion to represent this meausure. In particular we study weak convergence of a general class of locally Lipschitz functions of the log permeability, and apply this general theory to estimate errors in the posterior mean of the pressure and the pressure covariance, under refinement of the finite-dimensional Karhunen–Loève truncation.

Lower semicontinuity of weak supersolutions to nonlinear parabolic equations

We prove that weak supersolutions to equations similar to the evolutionary $p$-Laplace equation have lower semicontinuous representatives. The proof avoids the use of Harnack's inequality and, in particular, the use of parabolic BMO. Moreover, the result gives a new point of view to approaching the continuity of the solutions to a second-order partial differential equation in divergence form.