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A counterexample to Schauder estimates for elliptic operators with unbounded coefficients

Enrico Priola (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space R + 2 of R 2 . We show that for a particular initial datum, which is Lipschitz continuous and bounded on R + 2 , the second derivative of the classical solution is not uniformly continuous on R + 2 . In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.

A localized orthogonal decomposition method for semi-linear elliptic problems

Patrick Henning, Axel Målqvist, Daniel Peterseim (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this paper we propose and analyze a localized orthogonal decomposition (LOD) method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. This Galerkin-type method is based on a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear fine-scale computations on small patches that have a diameter of order H | log (H) | where H is the coarse mesh size. Without any assumptions...

A Multiscale Model Reduction Method for Partial Differential Equations

Maolin Ci, Thomas Y. Hou, Zuoqiang Shi (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a multiscale model reduction method for partial differential equations. The main purpose of this method is to derive an effective equation for multiscale problems without scale separation. An essential ingredient of our method is to decompose the harmonic coordinates into a smooth part and a highly oscillatory part so that the smooth part is invertible and the highly oscillatory part is small. Such a decomposition plays a key role in our construction of the effective equation. We show...

A note on the Rellich formula in Lipschitz domains.

Alano Ancona (1998)

Publicacions Matemàtiques

Let L be a symmetric second order uniformly elliptic operator in divergence form acting in a bounded Lipschitz domain ­Ω of RN and having Lipschitz coefficients in Ω­. It is shown that the Rellich formula with respect to Ω­ and L extends to all functions in the domain D = {u ∈ H01(Ω­); L(u) ∈ L2(­Ω)} of L. This answers a question of A. Chaïra and G. Lebeau.

A posteriori error estimates for elliptic problems with Dirac measure terms in weighted spaces

Juan Pablo Agnelli, Eduardo M. Garau, Pedro Morin (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...

A symmetry problem

A. G. Ramm (2007)

Annales Polonici Mathematici

Consider the Newtonian potential of a homogeneous bounded body D ⊂ ℝ³ with known constant density and connected complement. If this potential equals c/|x| in a neighborhood of infinity, where c>0 is a constant, then the body is a ball. This known result is now proved by a different simple method. The method can be applied to other problems.

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

Ryan Hynd (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We study the partial differential equation         max{Lu − f, H(Du)} = 0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution...

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