We prove partial and full boundary regularity for manifold constrained p(x)-harmonic maps. Contents

We prove global $$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) -regularity for minimisers of convex functionals of the form $${\mathscr {F}}(u)=\int _{\varOmega } F(x,Du)\,{\mathrm{d}}x$$ F ( u ) = ∫ Ω F ( x , D u ) d x .$$W^{1,q}({\varOmega },{\mathbb {R}}^m)$$ W 1 , q ( Ω , R m ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and controlled (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n .

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.

We prove global $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) -regularity for minimisers of $${{\mathscr {F}}(u)=\int _\Omega F(x,Du)\,{\mathrm{d}}x}$$ F ( u ) = ∫ Ω F ( x , D u ) d x satisfying $$u\ge \psi $$ u ≥ ψ for a given Sobolev obstacle $$\psi $$ ψ . $$W^{1,q}(\Omega ,{\mathbb {R}}^N)$$ W 1 , q ( Ω , R N ) regularity is also proven for minimisers of the associated relaxed functional. Our main assumptions on F(x, z) are a uniform $$\alpha $$ α -Hölder continuity assumption in x and natural (p, q)-growth conditions in z with $$q<\frac{(n+\alpha )p}{n}$$ q < ( n + α ) p n . In the autonomous case $$F\equiv F(z)$$ F ≡ F ( z ) we can improve the gap to $$q<\min \left( \frac{np}{n-1},p+1\right) $$ q < min np n - 1 , p + 1 , a result new even in the unconstrained case.

We consider the class of planar maps with Jacobian prescribed to be a fixed radially symmetric function f and which, moreover, fixes the boundary of a ball; we then study maps which minimise the 2p-Dirichlet energy in this class. We find a quantity $$\lambda [f]$$ λ [ f ] which controls the symmetry, uniqueness and regularity of minimisers: if $$\lambda [f]\le 1$$ λ [ f ] ≤ 1 then minimisers are symmetric and unique; if $$\lambda [f]$$ λ [ f ] is large but finite then there may be uncountably many minimisers, none of which is symmetric, although all of them have optimal regularity; if $$\lambda [f]$$ λ [ f ] is infinite then generically minimisers have lower regularity. In particular, this result gives a negative answer to a question of Hélein (Ann. Inst. H. Poincaré Anal. Non Linéaire 11(3):275–296, 1994). Some of our results also extend to the setting where the ball is replaced by $${\mathbb {R}}^2$$ R 2 and boundary conditions are not prescribed.