6 papers
Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs
André Guerra, Lukas Koch, Sauli Lindberg
For a nonlinear operator satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) is surjective, ii) is open at z…
The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces
André Guerra, Lukas Koch, Sauli Lindberg
We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, , where is integrable and bounded away from zero. In…
On the necessity of the constant rank condition for estimates
André Guerra, Bogdan Raiţă
We consider a generalization of the elliptic -estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura App…
Numerical evidence towards a positive answer to Morrey's problem
André Guerra, Rita Teixeira da Costa
We report on numerical experiments suggesting that rank-one convexity imples quasiconvexity in the planar case. We give a simple heuristic explanation of our findings.
Remarks on Ornstein's non-inequality in
Daniel Faraco, André Guerra
We give a very concise proof of Ornstein's non-inequality for first- and second-order operators in two dimensions. The proof just needs a two-dimensional laminate supported o…
Extremal rank-one convex integrands and a conjecture of Šverák
André Guerra
We show that in order to decide whether a given probability measure is laminate it is enough to verify Jensen's inequality in the class of extremal non-negative rank-one convex int…