Optimal Poisson Kernel Regularity for Elliptic Operators with Hölder Continuous Coefficients in Vanishing Chord-Arc Domains

REGULARITY AND FREE BOUNDARY REGULARITY FOR THE p-LAPLACE OPERATOR IN REIFENBERG FLAT AND AHLFORS REGULAR DOMAINS

Let ω(·) = ω(·, x) denote the harmonic measure associated to the Laplace operator and defined with respect to Ω and x ∈ Ω. A classical result concerning the harmonic measure, due to Lavrentiev [22], states that if Ω ⊂ R is a chord arc domain, then ω is mutually absolutely continuous with respect to σ, i.e., dω = kdσ, where k is the associated Poisson kernel. Moreover, Lavrentiev [22] proved tha...

L^p-Regularity for Elliptic Operators with Unbounded Coefficients

Boundary regularity for elliptic problems with continuous coefficients

We consider weak solutions of second order nonlinear elliptic systems in divergence form or of quasi-convex variational integrals with continuous coefficients under superquadratic growth conditions. Via the method of A-harmonic approximation we give a characterization of regular boundary points using and extending some new techniques recently developed by M. Foss & G. Mingione in [15].

improved regularity of harmonic map flows with hölder continuous energy ∗

For a smooth harmonic map flow u : M× [0, T ) → N with blow-up as t ↑ T , it has been asked ([6], [5], [7]) whether the weak limit u(T ) : M→ N is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function E(u(·)) : [0, T ) → R, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0, T ]. Here we show that if this exten...

The Regularity Problem for Second Order Elliptic Operators with Complex-valued Bounded Measurable Coefficients

The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with t-independent complex bounded measurable coefficients (t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in Lp ′ , subject to the square function and non-tangential maximal function estimates, if ...