Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

GLC Project I “Corner Singularities and Analytic Regularity for Linear Elliptic Systems” Part I: Smooth domains

Optimal 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...

Elliptic regularity and solvability for PDEs with Colombeau coefficients

The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of G∞-regularity of...

The Solvability of Concave-Convex Quasilinear Elliptic Systems Involving $p$-Laplacian and Critical Sobolev Exponent

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients

For functions from the Sobolev space Hs(Ω), 1 2 < s < 3 2 , definitions of non-unique generalised and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, wh...