Prescribed mean curvature equation on torus

Prescribed mean curvature problems on the torus have been considered in one dimension. In this paper, we prove the existence of a graph on the n-dimensional torus 𝕋 n {\mathbb{T}^{n}} , the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.

Rigidity and trace properties of divergence-measure vector fields

We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where {\varphi(t)} is a non-negative convex function vanishing only at {t=0}. We show that this property is always satisfied in dimension {n=2}, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when {\varphi(t)=ct^{2}}) in dimension {n\geq 4}. The validity of the quadratic rigidity, which we prove in dimension {n=2}, implies the existence of the trace of a divergence-measure vector field ξ on an {\mathcal{H}^{1}}-rectifiable set S, as soon as its weak normal trace {[\xi\cdot\nu_{S}]} is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.

Sharp solvability criteria for Dirichlet problems of mean curvature type in Riemannian manifolds: non-existence results

It is well known that the Serrin condition is a necessary condition for the solvability of the Dirichlet problem for the prescribed mean curvature equation in bounded domains of $${{\,\mathrm{\mathbb {R}}\,}}^n$$Rn with certain regularity. In this paper we investigate the sharpness of the Serrin condition for the vertical mean curvature equation in the product $$ M^n \times {{\,\mathrm{\mathbb {R}}\,}}$$Mn×R. Precisely, given a $$\mathscr {C}^2$$C2 bounded domain $$\Omega $$Ω in M and a function $$ H = H (x, z) $$H=H(x,z) continuous in $$\overline{\Omega }\times {{\,\mathrm{\mathbb {R}}\,}}$$Ω¯×R and non-decreasing in the variable z, we prove that the strong Serrin condition$$(n-1)\mathcal {H}_{\partial \Omega }(y)\ge n\sup \limits _{z\in {{\,\mathrm{\mathbb {R}}\,}}}\left| H(y,z) \right| \ \forall \ y\in \partial \Omega $$(n-1)H∂Ω(y)≥nsupz∈RH(y,z)∀y∈∂Ω, is a necessary condition for the solvability of the Dirichlet problem in a large class of Riemannian manifolds within which are the Hadamard manifolds and manifolds whose sectional curvatures are bounded above by a positive constant. As a consequence of our results we deduce Jenkins–Serrin and Serrin type sharp solvability criteria.

Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space

In this paper, bifurcation diagrams and exact multiplicity of positive solution are obtained for the one-dimensional prescribed mean curvature equation in Minkowski space in the form of [Formula: see text] where [Formula: see text] is a bifurcation parameter, [Formula: see text], the radius of the one-dimensional ball [Formula: see text], is an evolution parameter. Moreover, we make a comparison between the bifurcation diagram of one-dimensional prescribed mean curvature equation in Euclid space and Minkowski space. Our methods are based on a detailed analysis of time maps.

Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.

A generalization of the asymptotic behavior of Palais-Smale sequences on a manifold with boundary

In this paper, we study the asymptotic behavior of Palais-Smale sequences associated with the prescribed mean curvature equation on a compact manifold with boundary. We prove that every such sequence converges to a solution of the associated equation plus finitely many “bubbles” obtained by rescaling fundamental solutions of the corresponding Euclidean boundary value problem.