The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

We study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

DISTINCT SOLUTIONS TO GENERATED JACOBIAN EQUATIONS CANNOT INTERSECT

We prove that if two $C^{1,1}(\unicode[STIX]{x1D6FA})$ solutions of the second boundary value problem for the generated Jacobian equation intersect in $\unicode[STIX]{x1D6FA}$ then they are the same solution. In addition, we extend this result to $C^{2}(\overline{\unicode[STIX]{x1D6FA}})$ solutions intersecting on the boundary, via an additional convexity condition on the target domain.

On the jacobian equation J(f, g) = 0 for Polynomials in k[x, y]

Let k[x, y] be the ring of polynomials in two variables over a field k of characteristic zero.If f, g ∈ k[x, y] then we write f ~ g in the case where f = ag, for some a ∈ k = k\{0}, and we denote by [f, g] the jacobian of (f, g), that is, [f, g] = fxgy - fygx