An irregular complex valued solution to a scalar uniformly elliptic equation

Frehse, Jens
November 2008
Calculus of Variations & Partial Differential Equations;Nov2008, Vol. 33 Issue 3, p263
Academic Journal
We present an irregular weak solution $${u : \mathbb {R}^n \to \mathbb {C}}$$ of a uniformly elliptic scalar equation in divergence form with measurable coefficients. The solution has a square integrable gradient. Such examples have been known for dimension n = 5 only.


Related Articles

  • Some Properties of the Set of Homogeneous Solutions of Elasticity Theory. Sebryakov, G. G.; Kovalenko, M. D.; Tsybin, N. N. // Doklady Physics;Jan2003, Vol. 48 Issue 1, p42 

    Studies the basic properties of the characteristic set of homogeneous solutions of elasticity theory. Biorthogonal relation constructed for homogeneous solutions; Use of the Mittag-Leffler expansion for the meromorphic function in the integrand.

  • Homogenization of Periodic Systems with Large Potentials. Allaire, GrĂ©goire; Capdebosco, Yves; Piatnitski, Andrey; Siess, Vincent; Vanninathan, M. // Archive for Rational Mechanics & Analysis;Nov2004, Vol. 174 Issue 2, p179 

    We consider the homogenization of a system of second-order equations with a large potential in a periodic medium. Denoting byethe period, the potential is scaled ase-2. Under a generic assumption on the spectral properties of the associated cell problem, we prove that the solution can be...

  • Construction and practical use of two-scaled extensions for rapidly oscillating functions. Laptev, V. D. // Journal of Mathematical Sciences;Apr2009, Vol. 158 Issue 2, p211 

    This paper suggests several methods for constructing two-scaled extensions for arbitrary rapidly oscillating functions. Special attention is paid to the continuity (smoothness) of extensions. The author discusses the scheme of applications of extensions to homogenization problems in non-periodic...

  • HOMOGENEOUS LANCHESTER EQUATIONS.  // Encyclopedia of Operations Research & Management Science;2001, p369 

    An encyclopedia entry about "homogeneous Lanchester equations" is presented. These equations have one equation for each side and are used when the weapons for each side are homogeneous. It may also be a simplified approximation of a heterogeneous situation.

  • Scale-integration and scale-disintegration in nonlinear homogenization. Visintin, Augusto // Calculus of Variations & Partial Differential Equations;Dec2009, Vol. 36 Issue 4, p565 

    This work is devoted to scale transformations of stationary nonlinear problems. A class of coarse-scale problems is first derived by integrating a family of two-scale minimization problems ( scale-integration), in presence of appropriate orthogonality conditions. The equivalence between the two...

  • Plasmon analysis and homogenization in plane layered photonic crystals and hyperbolic metamaterials. Davidovich, M. // Journal of Experimental & Theoretical Physics;Dec2016, Vol. 123 Issue 6, p928 

    Dispersion equations are obtained and analysis and homogenization are carried out in periodic and quasiperiodic plane layered structures consisting of alternating dielectric layers, metal and dielectric layers, as well as graphene sheets and dielectric (SiO) layers. Situations are considered...

  • The disease of cultural homogenization. Golladay, Richard // Burnet Bulletin (Texas);4/2/2014, Vol. 141 Issue 14, p6A 

    In this article the author discusses the disease of cultural homogenization.

  • Continuum limits of discrete thin films with superlinear growth densities. Alicandro, Roberto; Braides, Andrea; Cicalese, Marco // Calculus of Variations & Partial Differential Equations;Nov2008, Vol. 33 Issue 3, p267 

    We provide a rigorous derivation by G-convergence of an effective theory of thin films in hyperelastic regime in the so-called discrete to continuous framework. By considering a discrete thin film obtained piling up at microscopic distance a finite number M of copies of a discrete monolayer ?,...

  • Homogenization of a class of nonlinear variational inequalities with applications in fluid film flow. Lukkassen, Dag; Meidell, Annette; Wall, Peter // Chinese Annals of Mathematics;May2011, Vol. 32 Issue 3, p417 

    The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure...


Read the Article


Sign out of this library

Other Topics