We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO(Ω)», a class neither containing, nor contained in\(C^0 \left( {\bar \Omega } \right)\). We also prove local and global Lp regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.

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On BMO regularity for linear elliptic systems

Introduction. 1. Preliminaries. 2. Integral inequalities. 3. The Laplace operator. 4. Strong solutions of the Dirichlet problem for linear equations. 5. The Dirichlet problem for elliptic linear. divergent equations in a nonsmooth domain. 6. The Dirichlet problem for semilinear equations in a conical domain. 7. Strong solutions of the Dirichlet problem for nondivergence quasilinear equations. 8. Weak solutions of the Dirichlet problem for elliptic divergence form quasilinear equations. 9. The behavior of weak solutions to the boundary value problems for elliptic quasilinear equations with triple degeneration in a neighborhood of a boundary edge. 10. Sharp estimates of solutions to the Robin. boundary value problem for elliptic non divergence second order equations in a neighborhood of the conical point. Bibliography. Notation Index. Index.

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Elliptic boundary value problems of second order in piecewise smooth domains

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.

Schauder Estimates for Elliptic and Parabolic Equations

We show that any weak solution to elliptic equations in divergence form is continuously differentiable provided that the modulus of continuity of coefficients in the L1-mean sense satisfies the Dini condition. This in particular answers a question recently raised by Yanyan Li and allows us to improve a result of Haim Brezis. We also prove a weak type-(1,1) estimate under a stronger assumption on the modulus of continuity. The corresponding results for nondivergence form equations are also established.

On C1, C2, and weak type-(1,1) estimates for linear elliptic operators

We estimate the size of the singular set of solutions to non-linear elliptic systems of the form where the vector field a satisfies a Dini-type continuity condition with respect to the variables (x, u).

Charles C Burch

Papers

The Dini condition and regularity of weak solutions of elliptic equations