Regularity versus singularity for weak solutions to elliptic systems in two dimensions
نویسنده
چکیده
In two dimensions every weak solution to a nonlinear elliptic system div a(x, u,Du) = 0 has Hölder continuous first derivatives provided that standard continuity, ellipticity and growth assumptions hold with a growth exponent p ≥ 2. We give an example showing that this result cannot be extended to the subquadratic case, i.e. that weak solutions are not necessarily continuous if 1 < p < 2.
منابع مشابه
Existence of at least three weak solutions for a quasilinear elliptic system
In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...
متن کاملOptimal Conditions for L-regularity and a Priori Estimates for Elliptic Systems, I: Two Components
In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the L-L-estimates, it yields the optimal L∞regularity conditions for the three well-known types of weak solutions: H 0 -solutions, L solutions and Lδ-solutions. Thanks to the linear theory in L p δ(Ω), it also yields the optimal conditions for a priori estimates for Lδ-solutions. B...
متن کاملExistence of three solutions for a class of quasilinear elliptic systems involving the $p(x)$-Laplace operator
The aim of this paper is to obtain three weak solutions for the Dirichlet quasilinear elliptic systems on a bonded domain. Our technical approach is based on the general three critical points theorem obtained by Ricceri.
متن کاملConvex Integration and the L Theory of Elliptic Equations
This paper deals with the L theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general L theory, developed in [1],[24] and [2], cann...
متن کاملOptimal Partial Regularity for Nonlinear Sub-elliptic Systems Related to Hörmander’s Vector Fields
This paper is concerned with partial regularity for weak solutions to nonlinear sub-elliptic systems related to Hörmander’s vector fields. The method of A-harmonic approximation introduced by Simon and developed by Duzaar and Grotowski is adapted to our context, and then a Caccioppoli-type inequality and partial regularity with optimal local Hölder exponent for gradients of weak solutions to th...
متن کامل